So caucuses are ran by the parties, and not by the State governments. That is why the two parties frequently don’t have caucuses on the same day in the same state–there is no special reason they should.
Since primary elections are both ran by and paid for by State governments, there is a major incentive to hold both party primaries on the same day–because it’s actually not super cheap (state budgets are usually tight, so even a couple million here or there is meaningful) to run an election, and to needlessly run two in the same year for the primaries in addition to the general election in November (which the States also pay to run) is an expense most states don’t want to deal with. As you see, South Carolina and maybe a couple other States have divided primaries, but they’re paying a price for it.
As for CarnalK’s question, everywhere I’ve lived that do primaries, they absolutely use the same polling places. There are different mechanisms to get a “Democrat” or a “Republican” ballot, and different rules about who is eligible for which and etc, but I’ve never seen them held on the same day but not have the same polling places.