Mileage seems to vary. I don't have any bot spam on Primal at all, meanwhile others in the AskNostr feed clearly have a ton on Primal. I don't really use Damus, so I can't speak to that, but others are saying they have no spam at all on Damus:

nostr:nevent1qvzqqqqqqypzpkwm5rs890dn200mqqsdu9v3y6h50e57zvl2jxaafr5tah3hxg8zqqsywz8hfpvc4cxs70g4qk4hzryhq7trm4h2d9wy0pqtzgrm8erexngtr5jul

For most clients, it's going to depend heavily on what relays you are reading from, not what client you are using.