How and why did e.g. Chechnya differ in status from entities like Armenia?

Constituent republics where subjects of Union Agreement. They have a constitutional right of secession. But all real power belonged to centralized Communist Party, so republican governments never had ability to implement this formal right. In general, division was considered by this reason, as formal and fictitious, and could be changed instantly at any moment, list of such changes is really very long.

At the end of Gorbachev's Perestroika Soviet Union was transformed from single-party state to multi-party system with democratic elections, it caused a dramatic shift of power from Communist Party to republican governments, and opened a door for dissolution. Because country suffered from really very serious economical crisis, governments of mostly economically developed Constituent republics decided that they will maintain better alone. Baltic states, Ukraine and Georgia also had strong separatist traditions.

It finished, when 3 most influential Constituent republics (Russia, Ukraine and Belarus) denounced Union Agreement. So, all 15 Constituent republics automatically became independent states.

It also meant, that all Autonomous republics, districts, and other autonomous entities stay DEPENDENT, because they were NOT subjects of Union Agreement, and never had a constitutional right of secession. Of course, many attempts were made to continue this wave of separations to the level of autonomous entities, but all of them considered as illegal by international community.

1. Chechnya at the moment of USSR collapse was part of united Chechnya - Ingushetia Autonomous republic. They unilaterally separated from Ingushetia and declared independence - unconstitutional by Russian laws.
2. Tatarstan Autonomous republic unilaterally declared itself Constituent republic in 1990. Russian federal government considered this act as unconstitutional, but luckily reached agreement with Tatarstan's authorities without war.
3. Abhazia Autonomous republic of Georgia unilaterally declared itself Constituent republic in 1990, and, after war, declared independence.
4. Southern Ossetia Autonomous district of Georgia unilaterally declared itself Autonomous republic (considered unconstitutional by Georgian laws), and, after war - independent.
5. Nagorny Karabah Autonomous district of Azerbaijan declared independence unilaterally.
6. Crimea at 1991 reached agreement with Ukrainian government about new status of Autonomous republic, at 1992 unsuccessfully declared independence.
7. Transnistria area of Moldova unilaterally declared itself Autonomous republic (never had autonomous status before), and later - independent.
8. Same attempt was unsuccessfully made by Gagauzia area of Moldova.

For now, considering autonomous entities of Russian Federation (former Russian SFSR of USSR), approximately half of them have Russian majority, and almost all of them (excluding Chechnya and Tatarstan) do not have, and never had before any notable separatist movements. Most of them are poor, and depend from financial aid from Russian federal government (for example, all autonomous republic of Northern Caucasus are sponsored by federal government).

Also Chechen wars show very clearly, that any attempts of declaration of independence can have consequences.

UPD

Other very important part is that Constituent republics have all formal attributes of statehood: national anthem, coat of arms, constitution and government. They have their own republican Communist parties (excluding Russian SFSR), branches of Communist Party of Soviet Union, that means high degree of REAL autonomy. Ukrainian and Bellorussian SSRs were member states and co-founders of UN.

At the times of foundation of USSR many people absolutely seriously considered Constituent republics as "independent" and told about, for example, foundation of Armenian SSR as "restoration of Armenian statehood". In real life, it was only facade for centralized single-party state. But constitutional right of secession always was a part of this image of Constituent republics as "independent".

USSR was designed not as a colonial empire of Russians, but a future world state that should be expanded all over the globe, so Ukrainian SSR will have same status, as future, for example, French SSR. Point of view that became dominated at 1930s, said that it is utopia, USSR never will be such world state, so it should stay within current borders, where traditionally dominated state is Russia.

And vice versa, Autonomous republics had their own constitutions, but didn't have symbols of statehood such as anthems, and never were considered as "independent".

Also, all decisions about changes of administrative division were made by Communist Party. In many cases, they even didn't follow formal procedures, and didn't made all necessary acts of republican authorities. It potentially create a minefield of territorial disputes.

"How" and "why" are two very different questions. Whereas the former can be answered formally, it is difficult to say "why" this happened other than "by historical accident". We'd need to delve into the history of each republic separately.

For some republics, the status changed both ways. For example, Karelia started off as an "autonomous republic" (ASSR) within RSFSR, then after the Winter War "promoted" to the full-fledged Soviet Republic, and then after "normalisation" of the relationship with Finland in 1956 demoted to the autonomous republic again.

Moldavia was once an ASSR within Ukrainian SSR.

Generally, either kind of republic had a traditional ethnic group with its own language, but they weren't necessarily a majority. For example, the Udmurt Republic never had more than about 1/3 population of Udmurt people (during the USSR time).

The Crimean Republic began as an autonomous republic within RSFSR, and then "demoted" to a region ("Oblast") after deportation of its main ethnic minority (and then changed hands and status several more times).

The Caucasian region went both ways. Georgia, Armenia and Azerbaijan formed their own federation and joined (or rather founded, together with others) the USSR. Other Caucasian peoples were initially lumped together as a separate Gorskaya ASSR within RSFSR, and then split and transformed multiple times, often alternating between ASSR and Oblast in status and in some cases forming "dual-ethnic" republics like Kabardino-Balkaria and Checheno-Ingushetia. (The reasons for such oddity deserve their own question).

So, the history is just complex. In the case of [Trans]Caucasus, the easiest simplification is that Georgia, Armenia and Azerbaijan were just bigger in population and had a history of their own statehood (even if long in the past).

The difference in status was somewhat formal, but it played important role during the dissolution of the USSR.

• Between the Soviet Republic and ASSR, the main difference was the level: the Soviet Republics formed the USSR (which was itself a federation), while ASSR formed (some) Soviet Republics (not only Russia/RSFSR had them).
• However, Soviet Republics had a right of secession, at least on paper, while ASSRs never had it. In 1991, Soviet republics could at least formally invoke this sleeping right, while ASSRs had no legal pretext to do it. This precipitated in several bitter conflicts (of course, as just one of the reasons), and not only in Russia (e.g. Chechen wars) but e.g. between Georgia and Abkhazia.
• Within SSRs (and RSFSR/Russia specifically), as well as on the USSR level, ASSRs had a bit more rights compared to other regions (such as Oblast). This was somewhat akin to states vs. districts in the USA or territories in Australia. For example, they elected more deputies to a separate chamber of the Soviet parliament ("soviet" is "parliament") resembling the Senate, where the number of representatives was determined per region rather than per capita. But since "soviet democracy" was a bit of a misnomer, most ordinary people didn't even know about it.