Just to add to what the above posters said, the biggest differences between Traveller: 2300 and 2300AD are matters of theme. GDW clarified the rules, gave you examples, and so on in 2300AD, but the biggest difference is the theme.
T:2300 was like "hey, here's a Traveller-like game set in an extrapolation of the Twilight: 2000 universe!" so you had nods to all of the typical Classic Traveller themes like trade, exploration, and so on.
2300AD has the tag-line "Man's Battle for the Stars" and as the name suggests, it's more narrowed down to a military-themed look. The Kafer invasion is a big deal, they got rid of that God-awful "comic book" style art they were fronting and replaced it with a grittier look in many cases. And they omitted the SK-19, though they still referred to the weapon.
---
As for the combat system, yeah, it has problems (the whole system has some huge problems - though GDW always had odd problems with their game systems having no reference to reality at some points in the case of 2300, if you want a good laugh, look at the throw range for characters though the ease with which you can hit something in the game is pretty good too), but I always liked it more than T2k's mostly because you could actually kill someone with a .22 pistol in a single shot.