In the case of LPII and LPIII, I used custom tilesets to make these packs stand out, as well as because I'd just put together a tool to make custom sets (in LPII's case) so wanted to make use of it, since I was unsure if anyone else would. For LPIII, it was also the fact that no existing tilesets at that time had the new objects. On the other hand, for LP Omega, I felt that using the LPII / LPIII sets would be good enough. For Doomsday Lemmings, it was again partly to use a new feature (non-palette-based tilesets), but more than anything I just didn't feel any existing tilesets fit the theme well enough - Dirt came close maybe, but not close enough for my liking.
I've never really tried making levels with
anyone else's custom tilesets yet, apart from a few levels back in the Cheapo days. Nowdays I'm pretty much only creating levels for NeoLemmix (and even there, I haven't created anything new, apart from one test level, since Doomsday Lemmings), and there are no custom styles for that yet apart from my own (and the Sega set if you count that one, though in some ways it's an official one); perhaps if some do arise at some point, I'll have a go with them.
When it comes to
playing levels, I'm not so fussed. You can make a good-looking level in almost any tileset if you try, and same goes for making a challenging and/or fun level; these things matter more than whether the tileset is custom or official IMO. Plus, since I don't tend to be a huge fan of Lemmini, there isnt' that much custom tileset content out there at the moment on Lemmix or NeoLemmix - though that very well may change with the release of NeoLemmix V1.29n and thus the possibility to convert a large portion of the existing Cheapo content over to it.