There's differences at all levels: where do I start?
From a security level: IE is intermingled with system code (to the point where during the anti-competitive monopoly court cases Microsoft declared that the system depended on it being there). This means that it has special access to certain parts of the system. If a cracker is able to get IE to crash, say, then they have a better chance of getting further into the system. Other browsers (Firefox and Opera, for example) are just normal applications: when they crash they crash. Have you ever noticed that when IE crashes, your normal explorer windows crash with it (and in Windows XP for example, your desktop window 'restarts')? That's because IE is tied to the system.
ActiveX: Another bastardisation of the internet that MS brought to the table was the idea of ActiveX. Originally, these were little programs that could be loaded off the internet and have complete access to and control of your computer. Naturally, only Good People(TM) were expected to use it. Naturally, lots of Bad People(TM) used it. Now MS have done quite well at patching these holes up but most IE exploits are related to IE being tricked into believing something is safe to run.
Security: Despite the reports to the contrary, MS takes longer (on average) to fix more critical (on average) security problems. People will point out 'yeah, well, Firefox had more holes found in X months', for example. These reports don't take into consideration that during those X months, IE still had critical holes that had been found before that time that still hadn't been fixed. That's all I have to say about that, as I don't want to get ranty.
User Interface / Usability: MS has done nothing for IE for how many years? Firefox and Opera have been working hard to improve the user interface and usability. Off the top of my head, tabbed browsing (instead of opening multiple windows, you open 'tabs' in the one window: keep your browsing contained!) and popup blockers are worth the change. The popup blocker stops unwanted popup ads without the need for any additional software. What are the two major things that MS seems to be bringing to the table with the next version of IE (when it finally arrives)? Tabbed browsing (which they boo-hoo-ed not too long ago) and a popup blocker.
Extensions: Now, I don't know about Opera here (though I hear they have quite a few extensions themselves) but Firefox has all sorts of extensions that fit easily into the application. The beauty of all of this is that you only install what you want. I have a little unobtrusive weather forecast in one corner. I have the ability to shake my mouse and close the window or have my RSS feeds display in a sidebar if I want. There's hundreds of different extensions available and they're relatively simple to write if you have a scratch you want itched.
Standards: Other browsers strive to be standards compliant. As IE hasn't been touched by MS for years they are being left behind, really. Basically the next version of IE will be catching up with most other browsers. There are heaps of sites out there that need to be broken to make it look right in IE: it's disgusting. The way I see it, the sooner a website realises that they don't need to continue spending time on getting it right for that broken browser, the sooner they can use that time more constructively.
Anyway, that's a start I think and I hope it was plain enough. The beauty of both Firefox and Opera is that they're free. Opera (www.opera.com) is free as in 'beer', Firefox (http://www.mozilla.org/products/firefox) is free as in 'speech' (we can get to that distinction somewhere further into the off season
SilentBob: have a play with the other browsers for a week each and let us know what you thought. I would truly be interested in your thoughts!