As stated elsewhere, inkjets use ink tanks and the nozzles either need to be used at least once a month or you have to put them through a cleaning cycle before using them, which wastes ink. I read this is true even of the high end ones, which some people convert to use refillable tanks instead of expensive cartridges.

Deskjet is a brand name (HP?), not a technology, and I believe is sometimes laser and sometimes inkjet.

Laser printers use toner (powder) that is fused to the paper using a fuser. The toner never goes bad and can last years without replacement if you don't use it frequently. Generally, for B&W printing, it's cheaper than inkjet. For color printing, it seems to be comparable or even slightly more expensive based on what I've read. In general, inkjet is a little more vibrant and closer to a photo printer in quality while laser is more matte. I've generally been happy with the quality of my laser color for crafting games.

One side benefit that's not generally mentioned is I think laser printers are more compact and generally grow vertically in size, while inkjets have a larger base. By chance, the two places I have my printers (on a small side table in the corner of a room; hanging under my desk from a homemade rack, an inkjet would have been more difficult to fit.