Copics are indeed another animal. They are alcohol markers created for artists and have been around quite a while. Then the scrapbooking world found them and their popularity went through the roof. However, not a lot of people who bought them on a whim understand what makes them unique, and potentially, more difficult to use.
Copics should be used on paper designed for alcohol saturation. Stamp ink and line art pens have to able to withstand the alcohol too. There isn't a Copic marker for every color under the rainbow. Artists who use Copic markers usually layer colors with other markers and gray markers to create contrast of shades and shadows. Copics are re-fillable and have different nib styles that can be swapped out. Essentially, once you buy the marker body, all you ever need again is a refill and maybe a nib once every few years. The end caps are airtight and they can be stored upright or flat without damage to the interior ink core.
If you are just going to flat-color a stamped image and not mess around with the techniques to add contrast, shading, shadows, etc. it might be easier to use any el-cheapo marker and toss it when it runs dry. If you are a die-hard user of a particular cardstock from one company, and that cardstock isn't a good base for Copic markers, you might want to use a different kind of marker. If you have a need to be matchy-matchy with your cardstock and inks, Copics might not be for you.
If you do get into them, I strongly suggest starting with just a few and taking a class to learn different techniques for their use. There are also a ton of online tutorials from incredible artists who share their expertise. And check out the copic store online - they have specials from time to time that might make it easier on a tight budget.