Divergences are when Stan can't follow the Hamiltonian trajectory---the Hamiltonian, which should remain constant, diverges. As @bgoodri says, R-hat just measures that the within-chain means and variances match the cross-chain means and variances. It's possible for draws to have decent looking R-hat when chains have not fully explored the posterior. Divergences are an additional way of signaling that something may be going wrong.
If you only have a few, increase adapt_delta and potentially lower stepsize (the latter is just the init value, so it's not as critical if you're converging to the typical set in all chains already).