The two words are synonyms. Literally, "乏味" means insipid and "枯燥" means dull and dry, so they can replace each other in many cases. People usually use them together as "枯燥乏味" to reinforce that mood. Sometimes they are not exchangeable according to the context, for example, you won't say "他很枯燥" instead of ”他很乏味“, or say "乏味的练习" instead of "枯燥的练习". If you are not sure which word should be used, just put them together, and you won't be wrong - ”他很枯燥乏味“ and ”枯燥乏味的练习“ are both correct.

它两是近义词，可以互换

There is no need to tell the difference between the two words, just use them as the same meaning. For example, these sentences have the same meaning: 这堂课很乏味。 这堂课太枯燥了。 这堂课很枯燥乏味。 If more accurate, maybe: 乏味，means tedious, not interesting, too simple; 枯燥，like Americans say "dry", means boring.

其实没有这两个词没有太大区别，常常连在一起用，如：这里的生活枯燥乏味。但在口语中，枯燥用的比较多。

枯燥侧重于没有生气，比较呆板。乏味侧重于没有味道，比较单调，二者合起来用效果更好。