Everything is in the title. Is there any context where 表す is used and not 意味する and vice-versa ?
意味する directly translates to "to mean", and it can be used in pretty much any situation where you would use "to mean" in English. (意味 > meaning)
単語の意味を調べる。 : search for a word's meaning.
And there are times when it carries more of a nuance of "showing".
愛を意味するギリシャ語。 : Greek (words) that "show" (mean) love.
As you can see, it has a slight nuance of "to show" here. But the base meaning of the word ("to mean") doesn't change.
表す directly translates to "to express (show)" as chargerstriker pointed out. And it can be used when you want to say that you express some feelings, what an image expresses/shows to you etc... So any kind of sentences where you would use "express" in English really.
喜びを顔に表す。 : Show/Express happiness on face.
Like most Japanese words, 表す may carry a special nuance in some cases, and a lot of learners confuse it with 意味する because of its nuance "to mean". But the root meaning of the word remains "to express/show" and "to mean" is nothing more than an interpretation of our foreign brain because we don't process languages the same way since they don't have the same logic.
花言葉で黄色いバラが表すのは嫉妬だ。 : In 花言葉, what yellow roses express/show (mean) is jealousy.
So don't get caught up by all the different meanings and tell yourself that there is one root meaning that can leads to many nuances, but they're nothing more than derived meanings.
Sources for examples sentences: