Could it cause problems if I replace JEI with NEI? Like, are there quests in the quest book that rely on JEI being used, or are there other game mechanisms that depend on it?

I ask this because I rather liked NEI than JEI. It had features (like the filter category dropdown at the top) that I miss, and I haven't found settings to enable them, and so I'm thinking of replacing JEI for NEI for myself.

Also, why do you prefer JEI? What are it's advantages?

JEI appeared as a stop-gap because NEI hasn't been updated for ages since ChickenBones has stopped maintaining his mods.

Right now the entire modded scene has migrated over to JEI making NEI a relic of the past. It's now widely used and mods usually provide native support for it, populating it with custom recipes without the need of installing tens of third-party plugins.

JEI provides nearly all necessary NEI-like features, but if you feel like lacking something see the NEI Features chart.

While you're free to replace JEI with NEI (just like you are free to add any mods you want), keep in mind that many recipe tabs are likely to disappear.

So basically by replacing it I would lose more functionality than what I would get.

Thank you for letting me know. I was out of Minecraft modding for a few years, mostly because of this: after 1.7.10 too many mod devs announced that they won't continue development, and I didn't really like the options that replaced their mods. One such mod was Applied Energistics if I remember correctly, but I've seen that they're developing it again, and that it's in the modpack too :)

keep in mind that many recipe tabs are likely to disappear

Well, actually the user who have taken over the development of NEI has though of this problem. I just tried it out, and it actually wraps itself around JEI, so recipes are properly shown (along with the machines that can make them on the left), JEI features are working, and the NEI features are also there.
Yes, I disabled JEI before launching.

Here is a short video about how it looks like.