I came across these terms in Darwin's "Origin of Species" and I wasn't sure what the difference is.
Usually, Stabilizing selection is a concept that applies to a phenotypic trait while balancing selection is a concept that applies to a given locus.
Balancing selection can either be due to negative-frequency dependence selection or due to overdominance (=heterozygous advantage at a single locus).
What Darwin may have meant
Because Darwin didn't know about genes, he was necessarily not using the term balancing selection as I am using it. I can think of several more or less related pattern of selection that might fall withing the definitions for "stabilizing selection" and/or "balancing selection". Maybe he meant "frequency-dependent selection", "positive selection for an intermediate trait", "varying selection through time (temporally heterogenous environment)" and "varying selection through space (spatially heterogenous environment)". Eventually again, that might mean "positively correlated traits that undergo opposite selection pressure (or the opposite)" but that would definitely be surprising.