Generally.... The thicker the strings, the more 'body' you get from your sound.
It's hard to describe in words but the sound is 'meatier' and more full.
The drawback with thick strings is they're hard to bend. So if you do a lot of string bending you may have trouble.
Thinner strings are easier to play but require a little more finesse. You can't press down too hard or they'll go out of tune. And you have to play softer as well.
The major advantage is that you can bend the heck out of them and if you're proficient at playing, get a lot more expression out of your lead lines.
For things like tapping, they'll make it much easier to achieve.
For a compromise you can use a hybrid set of strings that has a heavier bottom end and a light top end (ie, Dean Markley Special Lights)
Those are the ones I usually use because I tend to play rather heavy on the lower strings, but want the flexibility to do tapping and flowing leads when I need to as well.