An ink-jet print is printed from a computer visual file, while a silkscreen tends to be more of a hands on process, where the graphic designer can directly manipulate the printing screen in many ways. That's not to say hand drawn images can't be beautifully rendered with the ink-jet process, for they can. It's just that with the ink-jet printer the final image is more removed from the hands-on approach.
The longevity of ink-jet prints varies widely, but if archival paper and inks are used, a print may last anywhere from 50 to a hundred years. Silkscreen inks also vary in their longevity, but it is usually possible to get an even longer lasting print from the silk screen process.
Ink-jet printers are easy to use at home, where you can obtain excellent prints without any clean-up. Silkscreen printing requires a more elaborate workspace, a greater investment in materials and a lengthy and involved clean-up process that might involve working with noxious fumes.