The usual stuff, CTDs and infinite loading screens. For now I managed to make them happen less often again but I'm not sure what caused them. It seems like either the mod "Real Wildlife" or the script "Helgenrandomsoundscript" caused it. But, of course, it could also be linked with the fact that Skyrims engine's is just too old as it already was outdated when Skyrim was released.
Hmmm...personally I get issues with Fort Dawnguard or Volkihar castle with it periodically lagging like crazy, or with the portal in The Forgotten Vale and it freezing on the loading screens, other than that the Legendary Edition has been running a thousand times better than the original version of Skyrim did.