A while ago someone asked if it mattered that a game was four player vs 3 player. I didn't really have an answer then. One thing that I realized (later) is that I take 'swingier' actions with more players.
[This idea is based I read many years ago in the Bridge World, but I can't find it online)].
Assume that your opponents will score in a normal distribution. Now say you could wave your hands and magically score the same each time. A bit above average (say, 1 standard deviation). In a (generic) two player game, you'd win fairly often. But as more players are added, you'd do poorly. Let's make this concrete. You always score 13 points, and your opponents roll 3d6. With one opponent, you like your odds. With 2 or 3, you are doing ok. Anyone who thinks they'll win against six opponents has never rolled up a D&D character.
Single-session Matchpoint bridge tournaments take this to an extreme (which is why the article was published). With dozens (or hundreds) of pairs, skill gets you so far, but swinging for the fences is useful too. (Longer games and other forms of scoring change that).
So consider a "New Sparta needs a military world" opening. Do you explore +5 or +1/+1? With more players, I feel more strongly that Explore +1/+1 is correct ... Even if the +5 gives me a good card 100% of the time, I'd rather take my chances getting a good card and some spare cash. With three opponents, the safe 35 points isn't as tempting as shooting for 45 (but often getting 20).
As Race is often 2-4 players, this isn't a huge effect ... yet.
With more players you can also swing for the fences by calling speculative trades (produces), and other tactics.
What else? With more players I'm slightly more worried about releasing good cards for opponents (since 4 players tend to go through the deck two full times, while less players usually peter out at 1.5 or so), but that's a minor concern compared to other issues.
I haven't really thought of other issues relating to # of players...