Speaking from a purely technical standpoint, these are the strengths and drawbacks of the two:
The standalone is installed on your local machine. You can directly access it from an icon on your desktop. It takes up more disk space, is subject to disk corruption/partial deletion and upgrading has to be initiated by you or the update program. It uses the exact amount of memory less that a browser window takes to run than the web version. You also have the option of defaulting to a specific server (but the danger of misconfiguring your setup). If you have a dial-up or a slow computer, this platform will seem to be quicker.
The web based requires you to download a smaller file (the .jar) when you click on "trade". Usually this is cached but when an update occurs, you have to take the time to download it then and there. You always have the latest version and don't risk having to be part of a mass upgrade (like with 3.0). It takes a bit more memory to load (That of having a browser open) but other than that it is the same as the installed version.
The standalone works perfectly fine if you install it properly and keep your system defragmented and make sure you locate problems that crash it and remove them from the system. The web based is a lot simpler and requires less maintenance.
The listed minimum requirements are a Pentium 3 450mhz processor, 128mb ram and at least a 33.6 dial up connection. Which means with a semi-modern computer and a healthy dial-up connection, either version should work fine. Bear in mind, all of the requirements are in relation to what else you are doing. If you're throttling your internet connection or your processor, you're going to need more bandwidth or a faster processor / more memory etc.
Be kind to your computers =)