There really is no difference between the two. An inkjet cartridge is just a ink tank with a pump attached. So, it never "stops working" it just runs out of ink to pump. The 3-color cartridges are just like 3 separate cartridges in one package. If one of the tanks empties, there's nothing for the pump to move. The other 2 will work just fine. So, your colors will appear very strangely.
If you can find a printer that uses separate cartridges for each color, that's the most economic route(assuming you have no interest in refilling the cartridges). But, typically, only higher-end printers tend to offer this. Printer companies these days make the vast majority of their money from the ink. That's why you can find printers out there where a 3-color cartridge costs more than the printer did.
I'd like to know how a tri-color inkjet cartridge works vs. the individual color inkjet cartridges. If one of the three colors in a tri-color cartridge runs out, does the entire cartridge quit working? Or does it continue to work until the cartridge is truly empty and just give you a bad mixture of colors?