How to explain the apparent discrepancy in the use of papyrus vs. clay tablets between Ancient Egypt and Mesopotamia?

Most certainly we see mainly survivorship bias. Other factors play a certain minor role, but the durability of clay tablets, especially when fired, are most important.

Cuneiform was developed on clay, is easy to write in/on clay, and clay is abundant in that region. Unlike papyrus, unbaked clay may be recycled on the spot (erased) and much easier and cheaper to produce than papyrus in the first place. Papyrus needs an invention of layering the strips to be used as a writing material, and the invention of ink flowing from the stylus, and thus came later than writing on (eventually) hard surfaces like clay.

But papyrus is a native species to the Mesopotamian region:

The ecoregion is a complex of shallow freshwater lakes, swamps, marshes, and seasonally inundated plains between the Tigris and Euphrates rivers. It includes huge permanent lakes of Haur al Hammar, the Central Marshes, and Haur al Hawizeh as well as more seasonal ‘ahrash’ forest of Populus and Tamarix on islands and banks of the great rivers (Evans 1994). The vegetation of the ecoregion is dominated by aquatic plants – Phragmites (reeds), Typha (rushes), and Cyperus (papyrus) (Stattersfield et al. 1998).

— WorldWideLife: Tigris-Euphrates alluvial salt marsh

What papyrus doesn't have is longterm durability, giving us a perfect example of probable survivorship bias when we assume that Mesopotamians just 'didn't do papyrus':

Although Egypt exported its writing material to other parts of the ancient world, few papyri from outside Egypt survive. Only the climate of Egypt and certain parts of Mesopotamia favors the preservation of papyri in the debris of ancient towns and cemeteries.

— From the world of Papyri

But it is of course nonsense to conclude that the material was never used in Mesopotamia. Depending on the timeframe, it is more than certain that it was used extensively in the later times, of say, at least the first millennium BCE:

— WP: Bas-relief from room XIII of the royal palace of Khorsabad: the sack of the temple of the god Haldi at Musasir by the Assyrian troops, scene from the eighth campaign of Sargon II, 714 BC. Copy of a bas-relief from the palace of Dur-Sharrukin (Khorsabad) by Eugène Flandin, Monuments de Ninive (1849).

What's going on in that scene?

In the top left we see an Assyrian official dictating to two scribes. One writes in clay. The other writes on papyrus or parchment.

A similar scene from Niniveh:

— British Museum 118882, Tiglath-Pileser III on campaign, ca 728BC

This is explained as:

For example, in Figure 0.1, from the palace at Nineveh of Tiglath Pileser III (ruled 745–727 bc), one of the most successful commanders who ever lived, a beardless eunuch on the left calls out a list of booty while the presumably Assyrian-speaking eunuch in the middle records the inventory in the contemporary Assyrian dialect by impressing cuneiform characters with a stylus into a waxed wood tablet. The presumably Assyrian-speaking eunuch scribe on the right makes a duplicate record (to prevent cheating?) by writing on a roll of papyrus or leather, certainly West Semitic Aramaic characters tied to Aramaic speech. The difference in writing medium, part of any writing tradition, accompanies a difference in script and “underlying language.”

— Barry B. Powell: "Writing. Theory and History of the Technology of Civilization", Wiley-Blackwell: Malden, Oxford, 2012.

This forms an often seen pattern.

There are many further images of Neo-Aasyrian scribes in the field, always in pairs. Typically a clean-shaven scribe holds a papyrus or parchment, next to a bearded one with a hinged woden writing board, which opened out to reveal a waxed writing surface that could easily be erased for ruse (though other combinations of scribes are also known). The scribes are more often depicted pointing their styluses at the plunder or bodies they are counting than actually using them to write with. […]

Whereas the Assyrian and Babylonian dialects of the Akkadian language were still recorded with the increasingly cumbersome and recondite cuneiform script on clay tablets, Aramaic used an alphabet of just twenty-two letters, written freehand with ink on parchment or papyrus — neither of which organic media survives at all in the archaeological conditions of Iraq. All three languages, however, were also written on waxed wooden or ivory writing boards, several specimens of which have survived, and for which there is ample documentation in the cuneiform record.

The durability, tradition, and tamper-proof qualities of tablets ensured their continuing use for legal documents and religious literature, whereas erasable writing boards and ink-based media were particularly suited to keeping accounts. Cuneiform tablets had to be inscribed while the clay was fresh; the new media, on the other hand, could be drawn up, corrected, and added to overtime, like modern-day ledgers. It is highly likely, then, that the vast majority of tabular accounts from the first millenium BCE perished long ago with the objects on which they were written.

— Eleanor Robson: "Mathematics in Ancient Iraq. A Social History", Princeton University Press: Princeton, Oxford, 2008.

The references to organic writing materials like papyrus on clay tablets, or clay seals formerly on now rotten away papyrus scrolls, are the most perfect proof for the long term consequences of using unsuitable writing tools, which, like phones, just don't stand the test of time for archival matters.

The handbook says:

To compensate for the deficiencies of clay tablets, writing boards (Akkadian lē’u) with erasable waxed surfaces were used alongside them from at least the 21st century bc (Steinkeller 2004), plus papyrus (Akkadian niāru) from the mid-second millennium and parchment or leather rolls (Akkadian giṭṭu, magallatu) from the early first millen- nium onwards (see Philippe Clancier in Chapter 35). Practically no such artefacts sur- vive—apart from a few now surfaceless Neo-Assyrian writing boards—although they are occasionally mentioned in tablets and sometimes depicted visually. We must never forget that cuneiform culture was only one literate culture amongst several in the ancient Near East, albeit the most longlived and prestigious. […]

Enveloping declines in popularity from the Old Babylonian period, while the practice of producing duplicate tablets increases in popularity. The old technology does not alto- gether disappear, however, and envelopes are still found around Neo-Assyrian and some Neo-Babylonian letters and administrative texts. An echo of enveloping is hypothesized in Late Babylonian contexts, where an inner roll of papyrus is sealed with clay then a duplicate outer roll wrapped around it (Invernizzi 2003).

For depictions of scribes we rely on Neo-Assyrian evidence, largely from the relief sculptures of palaces. While objects are depicted accurately, Assyrian composition is formalized and does not yield photo-realistic images. Scribes are shown operating in pairs; one holds a wooden writing board with wax-filled panels, the other holds a pen and writes on a roll of parchment or papyrus. The second scribe is thought to be writ- ing a parallel account in Aramaic—the everyday language of the time—presumably for a different purpose. An alternative hypothesis is that he is a kind of war artist, tak- ing notes on the exotic landscapes to inform the later carving of reliefs celebrating Assyrian victory in that campaign (see Reade 1981: 162). The cuneiform scribe regu- larly carries a writing board rather than a tablet. From a practical point of view, it would be easier for a scribe on campaign to operate with a writing board than to have to source and process clay in a strange environment, make a proper tablet, and inscribe long lists before the tablet dries. In the Tell Barsip paintings and a few reliefs, however, the cuneiform scribe holds a tablet; he also holds a stylus as long as those of scribes writing on boards. […]

All these texts are written in the Akkadian or Sumerian languages and in cuneiform script, sometimes supplemented by short Aramaic epigraphs in alphabetic script. Akkadian had been the main language of Babylonia for centuries but this was no longer true in the Hellenistic period. Since at least the 7th century bc the most com- mon language had been Aramaic. Persian had always had limited impact in Babylonia, while the extent to which the Greek language was used after Alexander’s conquest in 331 is still a matter of debate (Geller 1997; Westenholz 2007). But for us, the key aspect is not a language’s popularity but the writing materials used to record it, as this dic- tates its survival. Leather and papyrus do not survive for long in the Mesopotamian soil and therefore very little Aramaic language material has been preserved, despite the fact that we can safely assume that Aramaic was commonly spoken. What little remains is recorded on stone or clay, either on pottery or on pottery sherds (ostraca) or on clay tablets. But the latter were used as a writing material only in exceptional cases, and the explanation for this is simple: while the cuneiform script is ideally suited for writing on the soft clay surface, resulting in the survival of a wealth of cuneiform documents in the shape of virtually indestructible clay tablets, this is not the case for the alphabetic Aramaic and Greek scripts which are meant to be drawn with a pen. For these scripts, a flat surface to write on is far better suited than a soft one to incise into, and writing materials such as leather and papyrus were therefore much preferred to clay to record them. But not a single such document has survived. Both alphabetic and cuneiform scripts were also recorded on wax-covered writing- boards, which survive only rarely: while earlier examples are known, especially from Assyria (see below) we do not have any such object, in either alphabetic script, from the Hellenistic period.

The question of textual survival is crucial when studying Hellenistic Babylonia. The thousands of clay tablets unearthed in various Babylonian cities form a sizeable and impressive set of evidence, but this must not distract us from the fact that what survives is just a very minor part of the written output of this period.

Texts in Aramaic and Greek certainly formed the bulk of the original documentation. Further, using cuneiform script in the Hellenistic period was not a neutral act: each cuneiform text, whether legal or scholarly, was written in a very specific social, cultural, and even political context. The use of the cuneiform script was, in itself, a signifier of the members of the old urban notability of Babylonia (Beaulieu 2006): cuneiform texts focus on the concerns of this social group and were written by and for its members. Therefore, although working on the history of Hellenistic Babylonia with the cuneiform sources entails using almost all of the available local texts, in reality we are dealing with only a very small and very specific share of the documentation originally written in this period.

— Karen Radner & Eleanor Robson (eds): "The Oxford Handbook of Cuneiform Culture", Oxford University Press: Oxford, New York, 2011.

The climate itself indicates that Egypt and Mesopotamia are indeed quite comparable, and not just for papyrus plant 'growing wild'. The air is also mostly quite dry. But that is not just all that's needed for papyrus material to have survived until today. The 'desert environment' for papyrus finds is greatly enhanced if that is a stable environment, like for example for the old finds we have in the Qumran scrolls, or those 4500 years old ones from Wadi al-Jarf: ceramic storage containing the scrolls in a cave.

That clay tablets are indeed less commonly associated with Egypt must not cloud the evidence we have that the Egyptians also used clay tablets. Like in the Amarna letters, in the lingua france of the time, Akkadian, written in cuneiform letters:

No curse formula was ever inscribed on his [Tutankhamun's] tomb walls, although the media reported that such a text was found inscribed at the tomb’s entrance. This might have been due to confusion with another find in the tomb of the young king: magical bricks – clay tablets usually hidden within plastered niches cut in the walls of the burial chamber, whose curse texts taken from the Book of the Dead protect the tomb and the dead at the four cardinal points.

— Isabelle Regen: "Curses, Egypt", in: Andrew Erskine, Roger S. Bagnall, Kai Brodersen, Craige B. Champion & Sabine R. Huebner (eds): "The Encyclopedia of Ancient History", Wiley, 2013. doi

We also have to observe a language change, plus a media change and changing traditions. The latter were of course the slowest to change. With the displacement of everyday languages from for example Sumerian and Akkadian towards the much wider adoption of languages like Aramaic, we see a change that corresponds to the writings. Aramaic is more suitable for papyrus and leather than clay, with leather much more preferred material over papyrus. But equally perishable.

But what about the often-quoted letter from the state correspondence of Sargon II (r. 721–705 BCE) that makes it unequivocally clear that this king did not think Aramaic a suitable means of communication with him? This is part of an exchange with a correspondent in the southern Babylonian city of Ur, as documented in Sargon’s reply. After quoting his suggestion to write in Aramaic, the king demolishes the very idea:

If it is acceptable to the king, let me write and send my messages to the king on Aramaic documents’: Why would you not write and send me messages in Babylonian (Ak-ka-da-at-tu)? Really, the message which you write in it must be drawn up in this very manner – this is a fixed regulation!

Sargon’s final instructions of how to write a proper letter refer to the format of the very document in which they are written: a clay tablet of the typical letter shape in portrait format, inscribed in cuneiform script and the contemporary Neo-Babylonian language – but not Neo-Assyrian, as Sargon and the other Assyrian rulers never expected their Babylonian correspondents to address them in this way.

The Assyrian kings always wrote their own letters to Babylonian recipients in Babylonian, as is also the case with this specific letter. This practice should be seen as part of a wider Assyrian strategy in Babylonia that sought to restore and preserve ancient cultural traditions, and Sargon’s directive to his correspondent therefore must not be interpreted as a general assault against the use of Aramaic as an epistolary language but as a deliberate attempt to shore up of traditional Babylonian customs in Ur.

In an attempt to explain why there is considerably less cuneiform material available for the state correspondence of the 7th century BCE, Simo Parpola postulated that Sargon’s successors permitted their correspondents the use of Aramaic, and that the resulting leather documents did not survive.

— RAD

An alternative translation of that passage might read:

"If it is convenient for the king, I will have (my messages) written in an Aramaic scroll in Aramaic and brought to the king,"

to which the king replies,

"Why don't you write in an Akkadian document in cuneiform and have (your messages) brought to me (thus) bring?"

The text uses two different words for writing in Aramaic alphabet and writing in Akkadian cuneiform: seperu and sataru. Similarly, late Babylonian texts differentiate precisely between the cuneiform writer tupsarru and the alphabet writer sepiru. On the other hand, there were undoubtedly scribes who mastered both writing systems.

— MPS

— RAD — Karen Radner: "Diglossia and the Neo-Assyrian Empire’s Akkadian and Aramaic Text Production", in: Louis C. Jonker Angelika Berlejung Izak Cornelius (eds): "Multilingualism in Ancient Contexts. Perspectives from Ancient Near Eastern and Early Christian Contexts", African Sun Media, 2021. (p147–181, doi)

— MPS — Michael P. Streck: "Keilschrift und Alphabet", in: D. Borchers/F. Kammerzell/S. Weninger (ed.): "Hieroglyphen, Alphabete, Schriftreformen: Studien zu Multiliteralismus, Schriftwechsel und Orthographieneuregelungen = Lingua Aegyptia-Studia monographica 3", (2001) p77-97.