Until recently I assume they were synonymous 😅, Here you go to Uni immediatly after finishing HS.
Universities can be public or private. They are simply a collection of colleges. An example of a single college that is not part of a university would be a community college.
As for which is more prestigious, most colleges not part of a university generally only offer Associate’s degrees with some offering Bachelor’s degrees. Universities offer those along with graduate level degrees. You are correct that universities tend to be tougher to get into.
Past your first job, nobody really gives a shit where you went.