Quicken: Too Buggy.
Mint: Too Simple.
Really, it all comes down to that. Neither product can be considered a serious solution for personal finance. In the software vs. webware battle, neither platform wins. But you lose.
Quicken 2007 (the version I use; I read that 2008 is similar) suffers from all the worst possible flaws of desktop software: It's slow, loaded with features, and unreliable. I've been using Quicken for 17 years, and the product has been getting worse each year (there have been some good years, but Quicken sees to it that you can't stand pat when you get a good version). My version of Quicken crashes when I look at it cross-eyed, and that's a scary thing in a program that holds so much personal data in a format that no other product--except the only slightly less-hated Microsoft Money--can read.
In Quicken's favor is its feature set. No online product does everything it does: balance your checkbook, pay your bills, track your stocks (including stock options and other vehicles), calculate your mortgage payments, budget for your retirement, integrate with tax planning products, and so on. It has more features than anyone can use, but I continue to be surprised by the weird financial transactions it can handle. If only one could use it without flinching. It's enough to drive you away from software forever.
Mint (review) is not a full-fledged personal finance product, but people keep comparing it with Quicken. It's a simple--too simple--personal financial analysis tool. It will tell you where your cash is going and how to save money, but it doesn't handle stocks (see Cake Financial), bills, long-term planning, or taxes.
If you're 20 or 25 years old and have super-simple finances, and if you don't mind going to another site, like your bank's, to pay your bills (or if you do so with paper checks), then Mint can help you make sense of your money. But once you start to get a complicated grown-up financial life, it runs out of gas fast.
Mint would actually be a great counterpart to Quicken. It can tell you things about your spending that Quicken cannot. But it can't replace it. Too bad.