From what I've learned in my time at school/From my personal experience, you are correct to put a comma before "like" in the sentence you used as an example.
Yes, in your example, there is a difference if you were to remove that comma.
Your example (WITH comma): "While I was visiting the zoo, I saw many different animals, like zebras, giraffes, and more."
Your example (WITHOUT comma): "While I was visiting the zoo, I saw many different animals like zebras, giraffes, and more."
Without the comma before "like," it would mean that the zebras, giraffes, and more draw attention from the many different animals, or it would mean that the many different animals are pleased with the zebras, giraffes, and more.
I'm unable to come up with an example that contradicts this because I struggle with making stuff up, and I'm on a tight schedule. If someone else can come up with an opposing example, that would be great.
I wish you good fortune and good health.