You're pretty close.
If I approve OF something, I think it's a good idea. I have an opinion.
If I approve something (no "of") it means I have the power/the responsibility to say "yes" or "no" to something.
I need to get my brochures approved by my legal department before they are final. The lawyer can say "No, I can't approve this," and if so, we need to fix it before it can be used. On the other hand, sometimes my boss doesn't always approve of my choice of photos, but he won't say "You cannot use this" -- instead he says "I would prefer another choice."
If I recommend something to you, I think it's a good choice FOR you (purpose).
I recommend [that] you try Firefox.
I recommended Firefox to my friend, but he's had only trouble with it. I wish now I hadn't recommended it.
I recommended this restaurant to my friend. They loved it and said they would recommend it for a special night out.
TO whom. FOR what (purpose).
Interested in Language