These two words are very similar and in most cases can be used interchangeably. The differences are very subtle and depend on whether you are using the words as verbs or nouns. In general, you want to use “influence” when you are trying to change the behavior of a person or system. You will use “impact” when you are speaking about results. Here are some examples:
I would like to influence the members of the board to vote in my favor.
The impact of my lobbying was apparent in the vote.
How can we influence what is said?
What is the impact of that statement?