I recommend you do not do this. I don't recommend you do this.

They both have the same meaning. You can also remove the "to" before the "do" in both cases. "I recommend you not do this" and "I don't recommend you do this" In my opinion the second way of phrasing this would be more common, in the United States at least.

Well, the result is the same but here's the subtle difference... "I recommend you not do this." This is a formal command. This is the proper grammar for the phrase (look up "present simple subjunctive" to see how it all works). "I don't recommend you do this." This is more like a suggestion. "I don't recommend you to do this." This is actually different! The thing being recommended is "you", not the action. It sounds like the speaker is recommending people for a certain task.

Both have the same meaning, it is just another way to say that.

Same meaning, just two different ways of saying the same thing.