I have yet to find good information regarding the s-process, so I'll just talk about the r-process here.
The key difference between the two is the conditions under which they take place. The r-process takes place during supernovae, specifically those due to gravitational collapse. In a collapsing star, many of the protons and electrons are squeezed tightly together - so tightly, in fact, that a process called neutronization begins. The protons and electrons actually combine to form neutrons (and neutrinos). This creates a large neutron flux, and so there are now lots of neutrons available for the r-process.
None of this lasts incredibly long, though, because the "explosion" doesn't last long. It soon propels much of the outer material of the star out into space, and the remains become either a neutron star or a black hole. That said, an interesting follow-up might address the relationship between the r-process, neutronization and the formation of neutrons stars in more detail.
Sources:
https://en.wikipedia.org/wiki/R-process
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