I have found Firefox to be fast but two issues have arose with me.
1. In mail applications such as Hotmail, School email. Script errors appear and block access from necessary functions.
2. When streaming videos that use java and then watching Youtube which uses flash at the same time, Firefox crashes.
Strange, I've never had any problems with that sort of thing.
In general, I'm not sure there's really any difference worth speaking of for most people. The speed difference is something you see in test labs, not everyday browsing where it's connection speed, latency and the server you're connecting to that dominate. As for memory, maybe some browsers use more than absolutely necessary, but unless you're trying to run them on 30 year old pocket calculator it's not something you'll ever notice. I just checked the actual numbers - with two tabs open, Firefox uses 53MB, Chrome uses 37MB. If you can find a laptop, let alone a PC, with less than 2GB these days you're probably looking in a scrap heap, so the fact that the code may not be perfectly optimised for RAM use really is irrelevant.
There are only a couple of areas where differences actually matter. Compatibility is a big one for programmers, and is the main reason people tend to say IE sucks. Microsoft tend to refuse to adhere to any standards, so sites that actually expect standards to be followed can have problems on IE that they don't on other browsers. Of course, others are no perfect on that score either, it's just that IE has tended to be the worst.
Security can technically be a problem, with different browsers having differently vulnerabilities and being differently competent at actually fixing them. Again, this is an area IE has tended to be criticised. However, by far the biggest influence on security is the user/keyboard interface. If you're sensible you don't need a browser to do security for you. If you're not sensible then, no browser can protect you.
So by far the most important thing is simply how it looks and feels. I've been trying out Chrome recently, and I don't really like it. It's a very minimalist approach, with very little on the screen other than the page you're viewing, few options and very little customisation possible. If you just want a browser that works straight out of the box, it seems perfectly good, but if you want to be able to play around with how things work a bit more, go with Firefox.
