bash string equality

In bash, what's the difference, if any, between the equal and double equal test operators?

[[ "a" = "a" ]] && echo equal || echo not-equal
[[ "a" == "a" ]] && echo equal || echo not-equal
[[ "a" = "b" ]] && echo equal || echo not-equal
[[ "a" == "b" ]] && echo equal || echo not-equal

results in:

equal
equal
not-equal
not-equal