One place to really see the difference is by considering Universal Coalgebra (also known as F-coalgebras) (in contrast to Universal Algebra)Algebra). The structures considered by Universal Coalgebra are typically infinite, whereas those considered by Universal Algebra are finite. This is taken from a computer science angle. Universal Algebra is about data types and Universal Coalgebra is about systems. Initial algebras are correspond to least fixed points, whereas final coalgebras correspond to greatest fixed points.
A free coalgebra (in this setting) corresponds to the final coalgebra of some functor (although one would call it a cofree coalgebra). A general introduction to the kinds of coalgebras I'm talking about is Introduction to Coalgebra by Jiri Adamek. This paper provides a way of constructing the final coalgebras for a certain class of functors.