Order as released is:
Codename 47, Silent Assassin, Contracts, Blood Money, Absolution.
Codename 47 had flaws, it's ok. But I'd say you mostly play it for its culturo-historical relevance (which is not much). No mid-mission saves and too much linearity are its downfalls.
Silent Assassin is a huge upgrade in all respects, but its suspicion mechanics were a bit out of whack. Enemies would become suspicious of you despite disguises if you did the wrong things. In worst cases the wrong thing was casually walking next to one. In some missions it made sense (you're a big white bald guy, it's ok for it to stand out in some locales), but other times it was just way too overzealous with revealing you. Still it was miles above Codename 47. Good game.
Contracts improves upon Silent Assassin. Which is not much, but is all that's really required. Mainly the not so broken suspicion system and... That's about it. The same old Hitman formula, slightly more distilled to perfection than its predecessor.
I still haven't got up to Blood Money or Absolution, but from what I've read BM is "more of the same" in a good way. It does spice things up a bit with connecting the missions more, allowing you to actually spend your earned money on things. Could well be up there with Contracts, someone else would have to pitch in here.
Absolution, well, from what I've read and seen it's a bit more different from it's predecessor. I guess a Rainbow Six pre- and post- Vegas comparison would be apt here. It's not necessarily bad but for a series purist it's going to be profanity unmatched. Linearity, lack of stealth, etc. the usual things that reboots "get wrong". But again, someone with actual play experience through the whole series might be better to comment here.