The 2 phrases are almost identical but have very small differences in meaning. "Let me give you a tour" could be more general, for example, if you showed up at someone's office, "let me show you around this place." Whereas "let me give you the tour" is more familiar, almost a joking / sarcastic way of saying the same thing -- "let me show you my house." But you could use either sentence and they'd be understood basically the same way.

Let me give you a tour. This means the same thing as let me show you around / let me show you the house.