Both words can be used either as nouns or verbs. They only become near synonyms as nouns. (This can be confusing, because the verb that comes closest to the meaning of "influence" is "affect" rather than "effect", which means to bring about some result. "To influence" is "to have an influence on", and "to affect" is "to have an effect on".)
To the extent there is a slight difference of meaning between an influence and an effect, I would say the former has more of a bearing on the process by which something happens and the latter on the state of someone or something compared to how it was before. So you might say for example that a candidate's campaign influenced the election, but that the election will have an effect on [or "will affect"] the economy of the country. But it would still be clear if you reversed the word choices in those examples.