What they are good for:
A good fountain pen with good ink will give you the most consistent ink flow of any tool available.
Cartridge pens provide better line quality than felt tips, with good thick and thin. They also travel pretty well.
The roundhand nibs of cartridge pens tend to be fairly rigid. This may be a feature or a bug depending on your preference.
If you need to write a whole lot with the same nib width, and you will use the pen regularly, a cartridge pen is probably the best tool for the job.
What they are not good for:
These are not pens to buy on a budget. Note my first sentence about "good" pens with "good" ink; that means pricey. Cheap pens and ink may give you okay results, but they may also be made of fail. Buying ink in cartridges is an even more expensive proposition than in bottles, and refillable cartridges are not everyone's cup of tea, though I rather enjoy the process.
Good fountain pens also need to be bought from a specialty store. You cannot walk into Michaels or your local craft store and buy one. Chances are that you will find Sheaffer's and Manuscript, and believe me, they are made of DO NOT WANT.
Cartridge pens need to be cleaned regularly. If you use your pens at least every other day, you can get away with only cleaning them every couple of cartridge changes/refills, but if you only use it occasionally you should clean it after every use. Cleaning them can be pain in the tush. Leave them uncleaned for long enough, and you will probably need to replace the whole pen as many companies don't sell replacement parts individually anymore (though you can always check Ebay!).
As mentioned above, the roundhand nibs on calligraphy pens with cartridges tend to be pretty rigid. If you like a flexible pen or need to get variations based on pen pressure, this is the wrong tool for the job.
For my money, I hate grinding the nibs on cartridge pens, but your mileage may vary.
I find that the variety of nib widths offered by cartridge pens is much narrower than for dip pen nibs.
If you want to use lots and lots of colors, then you need to have lots and lots of pens, or be prepared to wait a long time between switches as you clean and dry your pen.
Ariela's Favorite Brands:
I am a loyal devotee of Rotring Art Pens, though they have been cutting down their calligraphy line of late. 0.6 nibs are no longer to be had for love nor money, and 0.9s are only available on Ebay once in a blue moon. I like Rotring because they give good inkflow for a fairly inexpensive pen (by fountain pen standards), and they disassemble completely, which makes cleaning them - and especially drying out the parts afterward - much easier than the ones that don't break down so fully.
Unfortunately, my next favorite brand, Osmiroid, is no longer in production, though there are plenty of them floating around eBay. It's a workhorse brand with good value for your money. In the debit column, their nib and feed don't separate from the grip, which makes them harder to clean. However, the barrels of Osmiroid pens are shorter than the Rotring pens, which means they feel better in my hand.