If we want to be very specific, "I'm running late" is usually used when you've already started the process of getting to your destination, whereas "I'm going to be late" (not "gonna") can be used long before you've begun getting there.

I agree with Darren and Hugh. They are essentially synonymous, but "gonna" (referring to "going to") is more the spoken rather than written word. That's one feature of English. It frequently isn't spoken the way it's written, so you need to be aware of that. Although you can say it (I'm going to be late), I think most people would just say "gonna be late" more often, because it's quicker and seems a little less tedious. In spoken English, you could even leave the "I'm" out. This is very common, and just say "gonna be late." Occasionally, you might encounter someone writing "gonna" but it would not be considered the standard language. German does the same thing by contacting some verbs with a personal pronoun, but writing it as such is considered quite colloquial. It's the same thing as in English, it's just that this sort of speaking style is bit more common in English. I can't speak for other languages, but I imagine there are other examples in other languages like this.

They are both basically the same. 'I'm running late' has a bit more feeling of 'movement' - looking at the situation now, e.g the train isn't coming, I haven't finished packing my suitcase etc. 'I'm going to be late' is stating a prediction re the near future. But again, they are both very similar. (I'm a native English speaker, and English teacher).

They both mean the same thing. The second one is very informal. It is ok if you were saying this to a friend, but I would rephrase it as "I am going to be late." if you were using it in a business context.