These phrases could be interpreted the same way, if referring to a casual meeting. I should, however, point out one distinct difference:
I met my friend: Often refers to a past introduction - as in, "I met my friend many years ago." This often refers to the original meeting/being introduced for the first time. It's clearer if the word "first" is expressly specified, e.g. "I first met my friend many years ago."
I met with my friend: Generally refers to a explicit collaboration, especially a private one - as in, I met with my friend "privately" to discuss the issue. (It would be common for the "with" to be omitted here, especially when the meeting implies traveling to meet the friend.)