Difference

They are synonymous.

Does: The love of a mother have the same meaning as a mother's love? Or does it mean someone's love for a mom?

Also the love of animals sounds like it means animals are very giving and loving. Is that what it means tho?

Anyone please?

Does: The love of a mother have the same meaning as a mother's love?-- As it stands, yes.
Or does it mean someone's love for a mom?-- As it stands, no.

Also the love of animals sounds like it means animals are very giving and loving. Is that what it means tho?- Not as it stands.

So why does:

The love of a mother means a mother's love. And The love of animals don't mean an animal's love?

I don't get it.

Simply that in isolation as you have presented them, the natural intepretation, the expected interpretation, is as I have suggested. Of course, with the proper context, they can also mean as you suggest.

A photo of my father: what do you suppose that is?

A photo of dad

If you are not interested in learning, then I am not interested in teaching, Jones.

Sorry, it means a picture or photograph I have of my dad or father. Right?

A photograph my Father is in?

My point is that when we read it in isolation, there is one interpretation that prevails, just as happens with your previous sentences. On the other hand, what about 'my father's photograph'? That. standing alone, could be either him in the photo or him as the owner of the photo.