Why would I choose full service carrier instead of a low-cost one?

€1000-2000 is not a “normal” fare for a short-haul flight within Europe, even with a legacy full-service airlines. Most of them will have cheap fares that are slightly above the price of low-cost airlines (if you book a non-flexible return ticket in advance). Even full fares for a flexible ticket booked at the last minute or business class tickets should be in the hundreds, and not the thousands of euros.

You can however find fares of €2000 and up but that’s best regarded as a glitch of the booking system resulting from very complicated way airline fares are constructed.

Very often this happens when looking for flights from an airline that does not operate them in the area at all and simply resells tickets from partners. In these cases, there are almost always better fares available. The funny thing is that one website will show you the crazy fare but a search engine will reveal more logical routings or perhaps even the very same flights for a lower price.

Finally, note that not all passengers on the plane are paying the point-to-point fare. Some of the short-haul flights you might have looked at are really feeder flights to bring long-haul passengers to a hub. When combined with an intercontinental flight, the short-haul flight can be essentially free and the whole ticket cheaper than a direct long-haul flight to your destination from your airport of departure. Again a counter-intuitive results of complex fare rules.

So nobody needs a reason to pay that much money. Mostly people just pay a small premium for all the reasons detailed in other answers (convenience of non-stop flights and better airports, hopes of a better service in case of irregular operations, points for loyalty programs), especially if they are not paying for the tickets themselves. And sometimes, the full-service airline can be the only way to reach your destination or even be cheaper than a low-cost airline.

You actually want to be somewhere on time. I had an Air France flight from Budapest to Paris which didn’t fly and I told the desk I needed to be in Paris next morning 10am and that’s it. They put me on a Lufthansa flight via Munich and I was there on time. Do you think a low cost would this? At best you can rebook for free at worst you get the money the EU laws prescribe.

In addition to jpatokal’s excellent answer, budget airlines sell you a pair of one-way tickets, whereas full-service carriers sell you a return ticket. This means that the budget airline has considerably lower responsibilities to you if things go wrong.

When you have a return ticket, the airline has obligations to you from the moment you check in for the outbound flight: in particular, they’re obliged to get you home again. If the weather’s impossible on the day of your return flight, they’re must to put you up in a hotel and get you home as soon as they reasonably can. However, if you’re travelling with two one-way tickets, you’re out of luck. Your trip home is a separate journey, which doesn’t begin until you’ve checked in for that flight. If there are no flights today because of the weather, the airline can just cancel your flight and give you a refund, leaving you stranded in Whereversville at your own expense until you can find a flight home at whatever price they cost at short notice at a time when demand just went through the roof because everyone else is stranded, too.

Lots of reasons! Most of which boil down to prioritizing time and convenience over cost.

1. You need to fly to a city with no direct service. Most LCCs only sell “point to point” and will not cover missed connections, while full-service carriers do.
2. You have an unpredictable schedule and need the ability to change your flights easily. LCCs tend to fly less frequently and charge large arbitrary fees for changes, while full-service carriers operate high frequencies on their core routes and offer “flex” fares that have a high up-front cost, but allow unlimited free changes afterwards.
3. The low-cost carrier flies to some ridiculous airport in the middle of nowhere (eg. “Frankfurt”-Hahn, 125 km away), while the full-service carrier flies directly to the main airport.
4. It’s a long flight and you’re not willing to suffer for hours in a cramped seat. (Personally, I’ll fly in a sardine can for up to four hours, but beyond that I’ll at least fork out for a better seat.)
5. It’s business travel and somebody else is paying.
6. You want to collect frequent flyer points. (See also #5.)
7. You’re already an elite frequent flyer and want to use your perks (lounge access, priority boarding, etc.)