It’s confusing, but hopefully the fares map and table below should help clarify things.

The best overview I’ve found is from London Travel Watch’s guide to the Elizabeth Line:

How much does it cost to use the Elizabeth Line?

The fares on the central section opening on 24 May [2022] will be the same as
the fares on London Underground so for example, a single journey from
Paddington to Liverpool Street would cost £2.50.

Journeys to Heathrow include a £7.20 ‘Heathrow premium’ compared with
the same single journey on the Tube.

…

Mode of transport [Paddington to Heathrow]	Single fare (peak)	Single fare (off peak)	Journey time
Tube [Circle/H&C/District line, then Piccadilly Line]	£5.50	£3.50	56 mins
Elizabeth Line	£12.70	£10.70	26 mins
Heathrow Express	£25.00	£25.00	15 mins

Here’s the official, updated TFL downloadable PDF (the one embedded on their page is out of date) which shows the different fare zones (i.e. which part is the tube-like "central section" and which parts are priced like national rail).

However, this map seems arguably misleading because it doesn’t mention the special "Heathrow Premium". Really, I think they should say "Special Fares Apply" instead of "Fare Zone 6" for the Heathrow part. Here’s a screenshot with the missing Heathrow part annotated:

Experimenting a bit on Citymapper London (which shows many route options with price estimates), it looks like:

• the Heathrow Premium applies when you end a journey in a Heathrow Elizabeth Line station,
• the size of it depends on where you start your journey: e.g. Ealing Broadway to Heathrow is quoted as £6.60 via Elizabeth Line and £1.90 via District and Piccadilly lines, suggesting the premium drops to around £5.70 when starting outside of Zone 1.

As the Oyster/contactless system has expanded, fears of both revenue loss if fares are reduced and public backlash if fares are increased have led to the system becoming ever more of a mess.

Multiple stations with the same name, but for different modes of transport can attract different fares. Normally this is because different combinations of operators were assumed. Normally, for a particular zone combination, journeys that are assumed to use only TFL services are the cheapest, journeys assumed to use only rail services are more expensive and journeys assumed to use a mixture of rail and TFL services are most expensive.

Due to historic interavailability agreements and due to TFL taking over rail services, some services, including parts of the Elizabeth line are eligible to be counted in either the "rail" or "TFL" categories.

With that background in place, it’s time to talk about the Elizabeth line. My understanding is as follows.

From West Drayton to Abbey Wood and Brentwood, the Elizabeth line is priced as a TFL service on TFLs standard zonal fare scales. Some parts of it are also eligible for rail-only scales.

Shenfield in the east and stations beyond West Drayton in the west are outside the fare zones. From what I can tell when these stations were added to contactless pay as you go, the pay as you go single fares single fares were generally set by taking the price of a traditional rail return ticket and dividing it in two, while the caps were set based on the prices of out-of-boundary travelcards. This can lead to substantial jumps in fare at the edge of the zones, in some cases for journeys outside the zones it may be cheaper to forgo oyster/contactless and use traditional rail tickets instead. Particularly if you split ticket.

That leaves Heathrow as something of a special case. The rail (not tube) line to Heathrow is privately owned and has always attracted premium fares. When it first opened, the only trains were Heathrow Express. Some time later a service called "Heathrow connect" was launched as a collaboration between Heathrow express and GWR, the service was mainly intended for commuters to Heathrow Airport from the local area, but it also offered journeys to London at a price that was cheaper than Heathrow Express but more expensive than the tube.

As part of the Crossrail project the Heathrow connect services were taken over by TFL and were rebranded, first as "TFL rail" and then later as "Elizabeth line". Oyster, Contactless and travelcard acceptance were added. The Heathrow stations were placed in zone 6 for travelcard and capping purposes, but the single fares were set at a premium level rather than being set on the normal TFL scale. There is also no "out of station interchange" between rail and tube at heathrow.

To make things worse, the autocomplete functionality in TFLs single fare finder is rubbish and makes it very easy to select the wrong station. As I start typing "heathrow" into the search box, the underground stations appear when I reach "hea", the terminal 5 rail station appears when I type "heathr" and the remaining two rail stations don’t appear until I have typed "heathrow" in full.

The single fare finder is working perfectly in this case. When you type "Heathrow" it gives you options for the Tube stations at the various terminals, and the National Rail/Elizabeth Line stations at the various terminals.

If you click the tube station you see the prices that you would pay if you finished your journey at the various Heathrow Airport Tube stations and tapped out using the Tube station gatelines. But you didn’t do that — you used the Elizabeth Line station and so the system charged you accordingly.

The upshot is, be careful not only to select the right station but also the station with the right listed mode when looking up London fares.

As others have mentioned in the comments, the fare is much higher because the tunnel between Hayes and Harlington and Heathrow is infrastructure privately-owned by the airport itself, and they charge a reasonably hefty fee to use it.

One last point — though the TfL Single Fare Finder does work fine, I personally find it easier to use the unofficial site https://ltfares.com . Your mileage may vary though.