They can go together. I suggest Firefox for intense browsing and password saving (Home computer) and IE7 for multiple page surfing, and office work, not good for pass saving.
Here is what I found:
Names
Internet Explorer - Microsoft leaves no doubt what their products are. If they made toilet paper it would be called Butt Wiper. Internet Explorer isn't a bad name, but it's not spectacular.
Firefox - On the other end of the spectrum, the name Firefox gives no clue to what it is. Could be a car? 80s video game? Lame comic book superhero? Flaming dish at a wild game restaurant? A browser would probably be your last guess. No matter, it is a much cooler name than the utilitarian Internet Explorer.
Advantage - Firefox
Domain Names
Firefox - firefox.com Some kind soul donated the domain to the Mozilla Foundation. At least they own their own domain name.
Internet Explorer - internetexplorer.com Microsoft doesn't even own this one. It's one of those generic search portals masquerading as an IE site.
Advantage - Firefox
Mascots
Internet Explorer - The big blue e with a swirly around it. A nice simple graphic, but not necessarily a mascot. However, with Microsoft's history of mascots (Clippy, Bob), maybe it's better they just stick with a letter.
Firefox - A giant fox with his tail on fire attacking the earth. That's the stuff of nightmares. Not as scary as a communist Godzilla but close. It's a cool graphic but Internet Explorer has to win something doesn't it?
Advantage - Internet Explorer
Google Fights
Why does Internet Explorer crash all the time? - 2.57 million results
Vs.
Why does Firefox crash all the time? - 218,000 results
Next...
Internet Explorer - 22.8 million results
Vs.
Firefox - 23 million results
That one is definitely a surprise.
Next...
Internet Explorer will help me get laid - 1.34 million results
Vs.
Firefox will help me get laid - 75,700 results
A much better chance of getting lucky with IE.
Next...
Internet Explorer came with my computer - 6.9 million results
Vs.
Firefox came with my computer - 659,000 results
Tie breaker...
Internet Explorer is the best browser ever - 2,110,000 results
Vs.
Firefox is the best browser ever - 474,000 results
Advantage: Internet Explorer
Coin Flip - Best of 7
The coin flip was conducted using a 2000 Sacajawea dollar.
Internet Explorer - Heads
Firefox - Tails
Results: H-T-T-T-T-H-T
Advantage - Firefox
Stereotypical User
Internet Explorer - Brain dead newbie. Loves pop-ups, viruses, and spyware. Just wants to "surf the Internet and check my email." Oblivious to alternative lifestyles. Seeks help from "computer smart" nephew.
Firefox - Proselytizing ubergeek. Loves freedom, choice, and tabbed browsing. Just wants to "improve mankind with Open Source Software." Oblivious to market forces and the power of money. Seeks other geeks to join in on the evangelism.
Both are equally annoying.
Advantage - Even
Conclusions
It was a close battle, but Firefox edges out a victory against the Internet Explorer by a score of 3-2 with 1 tie and wins our Technical Award of Excellence.