The question is pretty much contained in the title:

What are examples of equivalence relations of topological spaces which are neither stronger nor weaker than homotopy equivalence?

Something that comes to mind is cobordism, for instance, but I was wondering about the existence of equivalence relations which are fairly different than homotopy (and that work for all topological spaces), or perhaps somewhat related to homotopy equivalence but very different in motivation.

I guess that this can be reduced to asking for equivalence relations on maps since homotopy equivalence of spaces is defined as equivalence of compositions of maps to identity maps.

I have no particular area of mathematics in mind for the origin or use of such equivalences, only that they work for all topological spaces.