k-means forces every point into exactly one cluster, with no notion of confidence. A Gaussian mixture model asks a softer question: what's the probability this point belongs to each cluster? Answering it requires an algorithm — expectation-maximisation — that recurs throughout probabilistic ML wherever there are hidden variables to infer.
k-means needs to be told $k$ in advance, and assumes every cluster is roughly round. This page covers two families that relax those assumptions in different directions: hierarchical clustering defers the choice of cluster count to a visual cut, and density-based clustering abandons the "round cluster" assumption entirely by defining a cluster as a connected region of high density.
k-means is the clustering algorithm everyone reaches for first — and its simplicity conceals just how strong its assumptions are. It assumes clusters are round, similarly sized, and similarly dense, and it will happily produce a confident, wrong answer when those assumptions don't hold.