As in title, in terms of group theory (I'm not familiar with category-theoretic terms), question comes from algebraic topology but seems to be of general interest. (Other questions on MSE touch on the topic but I haven't found a direct answer).
What is the difference between the two? Is one special case of the other?
In the category of groups, the free product with amalgamation is the realization of the pushout when both functions are embeddings. More generally, say that $f\colon H\to G$ and $g\colon H\to K$ is a pushout diagram. If $\mathrm{ker}(f)=\mathrm{ker}(g)$, then the pushout is given (up to unique isomorphism) by $G*_{f(H)\sim g(H)}K$.
However, there are pushouts that cannot be realized as the free amalgamated product. For example, if $H$ is nontrivial, $f$ is the trivial map, and $g$ is an embedding, then you can’t get the amalgamated free product, because you cannot amalgamate the trivial image of $H$ in $G$ with the nontrivial image of $H$ in $K$.