Calling it an exploit seems politically charged. There wasn't an exploit in the Ethereum software itself, which behaved exactly as intended. Instead, someone wrote a shitty, poorly-designed contract that had a loophole in it. Someone took advantage of that loophole and make a bunch of money off of it. To me, "exploit" has connotations of breaking the ethereum system somehow, which was not the case.
Ethereum Classic seems to be sticking by the original stated purpose of Ethereum (to have fully automated, non-repudiable contracts).
The other Ethereum seems to be instead stating that it's acceptable for the judgement of the Ethereum devs to have ultimate control and veto power over anything that happens on the Ethereum blockchain. In other words, Ethereum is more of a fiduciary contract system than any sort of "objective" contract platform.
To be clear, I have no vested interest either way. I have never been involved with Ethereum.