Progressive house is normaly quite melodic , almost similar to trance sometimes. Nick Warren springs to mind.
Minimal is as the name suggests ,made up of fewer elements , less melodic. lots of bleeps and glitchy , wonky sounds. Essentially a type of techno.
Tech house and Deep house now cover quite a broad range. At the techier end you have house and minimal sounds fused together sounding maybe futuristic as opposed to old school funk guitar sounds.
Deep house seems to mean alot of different things to different people so I'll let someone else chime in on that.
Bare in mind that all these sub genre names are abit silly and restrictive as 1 tune may have elements of different genres.
Maybe some examples would be helpful but inevitably someone will say "thats not progressive house" etc