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.)

They're really the same thing. I think perhaps at some point, some people stopped including the word "with" and it became understood that that was implied. However, in the stricted sense, using "with" is more grammatically correct.

meet: 用于中性的陈述。例如：我在大街上遇到了一个朋友，这个“遇到”一般都是中性的，有时甚至是较喜悦的。 meet with：遭遇。例如：我在大街上遇到了一个逼债的债主，这里的“遇到”是“遭遇或遭受“到“不好”的事或人。