Both seem to mean "to be helpful, to be useful" but I cannot distinguish what is their difference.

Here are some examples:

昨日の帰りに少し摘んどいたんだ‌ 具材の足しになるかな?

I gathered these on our way back yesterday.‌
Will these work as additional ingredients?‌


Even if this map does not seem useful, bring it with you.

足し is used almost exclusively in the idiom 足しになる/足しにする, but 足し on its own means something like "(small but nice) addition/complement". The idiom is used only in a situation where the amount of something is considered insufficient. Someone who says 具材の足しになるかな thinks there are not enough ingredients but the gathered plants will work as an addition. Also note that 足しになる also implies the added amount is not very large. It's fine to give someone money saying 学費の足しにしてください, but it can be inconsiderate to thank someone saying 学費の足しになります. We don't usually say この地図は足しになる because having more than one map rarely makes sense in the first place.

役に立つ ("to be helpful") can be used in a much wider variety of situations.

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