They're pretty different applications for different uses really. Photoshop is for pixel based editing and AI is for vector based editing. In design's I'll use both. If I need something that looks illustrated I'll use Illustrator then bring it into my PS document as a smart object.

There is no right or wrong answer. Id say I prefer using Photoshop, however, I've never played around enough with Illustrator to make that statement as valid as someone who has used both.

I'd never used illustrator until I had to design a logo and so I taught myself the basics.  It's a great program for making shapes and logo looking things.  I like how you can just click on an object and don't have to rummage through the layers like you do in photoshop.  However, just as everyone else has said, they are both good in their own right for different reasons.  Im glad that i've learnt illustrator as well because I can utilize the strengths of both programs depending on what im trying to achieve. I highly recommend becoming well acquainted with both!

As has been said above, the programs are completely different. Most people go with photoshop because it's more popular and it's the standard, but Illustrator actually has huge capabilities as well and you can do far more with Illustrator than most people would expect. I'd recommend learning your way around both programs so you can use each to the best advantage.