Is a spacetime of constant positive curvature just a 4-hypersphere? In discussions of basic cosmological models, I don't see "spacetime of constant positive curvature" described more simply as a "4-hypersphere".  What am I missing?  
 A: You are presumably thinking of the FLRW metric for a universe with greater than critical density i.e. a closed universe.
We normally use comoving coordinates to describe this, in which case the time coordinate is not curved and at every point along this time coordinate the three spatial coordinates have the topology of a 3-sphere. That is, if we draw a straight line in any direction and continue it indefinitely the line will eventually return to it's starting point.
This isn't a 4-sphere because this is not true of the time dimension. The time dimension starts at the Big Bang and ends at the Big Crunch so it is just a line not a loop. In fact it is geodesically incomplete at both ends since both the Big Bang and Big Crunch are singular points.
A: There are different descriptions of Spacetime according to General Relativity. 
Look at the De-Sitter-Space. It is a mathematical concept of Spacetime with a  positive curvature. It is a submanifold of Minkowski-Space. 
there is also an Anti-De-Sitter-Space, which has a negative curvature. It plays a role in some cosmological theories (like Inflation).
The curvature of Space is also addicted to the Hubble Constant and the measured density parameters. 
It can not be said much about the global topology of the Universe but it seems as if it is flat. 
