I agree that 言語 and 言葉 share the same meaning in many cases.
I feel that the difference between them is 言語 has a formal and academic connotation whereas 言葉 is an everyday word that anyone can use casually.
For example, a Japanese word that means linguistics is 言語学(げんごがく).
But we don't call linguistics 言葉学.
Similarly a Japanese word that means a scholar that specializes in linguistics is 言語学者(げんごがくしゃ).
But we don't call the scholar 言葉学者.
I think 言語 is a language system whereas 言葉 is each word that constitutes the system.
If 言語 is a whole tree, 言葉 are the leaves.
葉 is kanji that means leaf or leaves.