Dye sublimation produces very soft, almost "unfeelable" prints. It uses a transfer paper specific to dye sub. The ink, as well, is specific to dye sublimation. This can only be used (as far as I know) on white or light garments.
Inkjet transfers use inkjet transfer paper, and either dye ink (not dye-sub ink) or pigment ink, depending on the paper. They can be applied to dark AND white shirts using specific transfer paper for lights and specific transfer paper for darks. Inkjet transfers for lights have progressed to a very nice, light print quality as well (check out
jpss paper), but transfers for darks are still a bit thick on the shirt. You can use the same ink for lights and darks, given that the papers you use both accept the same type of ink. In my own experience, pigment ink for inkjet transfers works great and seems to last a bit longer than dye ink. I actually use a 1400 with a pigment CIS setup for my transfers.
EDIT: I believe that certain Epson models, like the 4880, can do pigment/dye AND dye sub printing. You'd have to do more research, I don't know much about it.