"On me" vs "I currently have"

Both phrases mean the same thing. They mean that if someone asked you for ten dollars you would be able to provide that. However the use of the adverb "currently" means that you don't always have ten dollars available. Maybe yesterday you had twenty, and the day before, no money. "Currently" talks about "in the present time," either in terms of clock time or in terms of what is happening in your life. "I am currently working in Paris," means that you are working there now, yesterday, and most likely, tomorrow. "He is currently in the lead," is a different kind of present and is probably confined to an hour or two or maybe seconds of time.

"On me" is a prepositional phrase, and another way to say "on your person." If someone came up to you asked you for ten dollars, you could reach into your pocket and produce the money. In the second phrase, it is not known whether the money is in a pocket or in your hand. You have it, but it may be at home or in the bank. Consider these variations: "I have ten dollars in my hand," or "I have ten dollars in my left pocket," or "I have ten dollars in my briefcase."