It is true that they both make sense without the word "over", but if I were to say "I'm glad if you come for dinner" there is a little bit of ambiguity about where dinner would be. Dinner could be at a restaurant, at a friends house, anything. Saying "I'm glad if you come over for dinner" most of the time implies it would be at the house of the person speaking, unless another place was specified earlier in the conversation.

We can also say 'Come round for dinner' , with exactly the same meaning.

In this example, "over" suggests a crossing from one place to another. From your place to mine.