Really depends what you're comparing;
For example, a budget 240mm vs a high end 120mm may not show substantial gains.
Similarly, a high end 240mm vs a budget 120mm will be night & day.
Generally, all else being (roughly) equal in terms of quality etc, a 240mm will show significant gains over a 120mm.
You're not talking double/half, it doesn't quite work like that..........but you should be looking at 10'C or so benefit in most cases, with a 240mm.
IMHO, there's really not much use for 120mm rads, outside of really tight mITX builds that cannot accommodate much of an air cooler.