Both have the straight feed ramp to reliably feed JHPs, but the MkIII has a loaded chamber indicator as well on the barrel. Different ejection port (changes on the MkIII to deal with cracking from shooting a bunch of SMG ammo, IIRC). Front sight on the MkII is integral to the slide as is the rib that it is a part of; it can't be changed. MkIII uses a dovetail (the classic used a staked and soldered sight). The MkII used the older, smaller rear dovetail but with a better sight than the classic. The MkIII rear sight is better. MkII has the extended bushing, the MkIII does not. Most MkIIIs have cast frames; all MkIIs are forged.
Most MkIII pistols are actually the MkIIIS. Those have a firing pin safety. The Israeli MkIII imports are usually the MkIII proper and don't have it. There is also a MkIIS, which is a MkII with the firing pin safety, which came out later during MkII production. Israeli MkIIIs also have a rib on the bottom of the frame.
There are also hybrids that have features of the classic, MkII, and/or MkIII. Some MkII and MkIII pistols were also made with older features, like the old-style safety, the ring hammer, lanyard rings, and the older style plastic or wood stocks.