What is the best engine for web and for mobile?

That question is impossible to answer easily, and most people have strong opinions on their preference. Both can work, although Unity is more commonly used on both those platforms.

Do you have any more specific requirements in mind?

2 hours ago, jbadams said:

Some relevant discussion. 

A update on that. Unreal is now only 40mb- 60mb larger for the exact same game as Unity.

I tested this with my own mobile games. Unreal has the better UI tools tools with better quality. Unity was easier to publish because wit Unreal you need to alert them that you want to published a game.

In short both Unity and Unreal are very good options for mobile games, Unreal is improving constantly and should soon be as small as Unity per game.

If you are a indie game programmer(or small team) like me. I suggest Unity. It's widely used, easy to learn, and its asset store is much more resourceful than UE's. And its supporting forum is superb.