The two phrases do have the same basic meaning. However, the first has more emphasis. "Once every week, I see my friend." In this sentence, the time and frequency are emphasized. "Every week I see my friend." In this sentence, the time and frequency are less important.
Note: this only works when it is "once." "Twice every week, I see my friend," does not put any special emphasis on the frequency. Also note that we tend to prefer having one part of the equation be singular. "Once every three years" works, and "three times every year" works, but "twice every three years" sounds very awkward.