There is no difference in the meaning but generally to have an effect is usually when we are for example reducing pain, or talking about something decreasing. Maybe something we do not want. Influence is much more general and can mean anything which changes something else (or someone else) in any way, good or bad.

Teacher Joseph's answer is a good answer. I just wanted to note that "effect" is more often used to describe an immediate change, but influence often implies long term or slower changes, even if it is negative. "The car accident had a bad effect on John; ever since then, he is afraid to drive" versus "My father had a negative influence on me; he made me become the type of person who is afraid to take risks"

It's what you are reading. See https://books.google.com/ngrams/graph?content=have+an+effect%2Cinfluence%2C+have+an+influence&year_start=1800&year_end=2008&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Chave%20an%20effect%3B%2Cc0%3B.t1%3B%2Cinfluence%3B%2Cc0%3B.t1%3B%2Chave%20an%20influence%3B%2Cc0

My immediate reaction is that in most cases these things are more or less synonymous, but there are some circumstances where you would use one and not the other. For example "I watched Titus Andronicus yesterday and it really had an effect on me." This means an emotional or thought-provoking effect; "influence" wouldn't be right here, even though it might influence you in the future. "That friend of his is a bad influence." You wouldn't say he "has a bad effect on him", which would (to my mind) imply a physical, probably medical, effect rather than a moral one. Those are just two examples. I daresay the examples you're talking about, though, there probably isn't any difference.