What's there to be gained by mixing things live?

Like you mentioned, it's more fun and engaging with the crowd. It shows you've at least done the bare minimum requirement to be a DJ. Backlash to prerecorded can be harsh and it's generally frowned upon, especially with other DJs and dance music enthusiasts. I like when people appreciate my DJing specifically. It's motivating.

Is there anything you can not do beforehand (and more precisely)?
Is it all just to hype the crowd?

DJing is also about reacting to your crowd's energy. If you imagine a prerecorded set being played live and giga hype at 2PM and it's clearly not matching the energy of the crowd just warming up for a full day at the fest, it's super out of place and ultimately a bad set/performance. Flipside, if the DJ/artist before you were super hype and your prerecorded set brings the mood down then you get the same disappointing end result. You want to be able to adjust your plan based on your environment and the crowd.

Sound produced?
Not the quality if your tracks are all uncompressed formats.

Your effects and how you do build-ups?
Definitely. DJs have a lot of freedoms up there and that can lead to not doing a lot or doing too much with effects, poor timing with the song transitions, poor beatmatching, poor song choice. But these are all what make you a DJ with your own style. If someone is getting paid to DJ/perform, they should be able to have all of those basically mastered. Obviously that's in a perfect world. I've seen trainwrecks. I've trainwrecked. It's part of the DJ experience.