What's the difference between the LG G6 model numbers h870, h870s, h870k and h870v?

Suffix variants are meant for different regions and carriers around the world, mostly for support of different regional LTE bands.

From PDAdb: H870 supports 14 bands; H870S supports 8; H870K only supports 5. I don't see a "V" variant listed (it's against the naming norm of LG, actually).

AFAIK, Europe uses LTE Band 20 primarily, so both H870 and H870S should do the job locally just fine, but considering international roaming capabilities, H870 is the one you want to get if you do have the money to shell out (EU variants are almost always the most expensive).

One more thing: if you want to tinker with the phone at all (even just rooting, even not immediately), buy the EU variant without hesitation. Only EU (H870) and US Open (US997) variants will ever get official bootloader unlock from LG - not even the Australian variant (which also bears H870 with no suffix).