運動をする and 運動する are the same in meaning. It's similar to "to do exercise" vs. "to exercise" in English. Grammatically, 運動 is a noun in 運動をする, so を is necessary when 運動 is modified, thus, for instance 腕の運動する is ungrammatical. (Actually, 腕の運動する is used in colloquial expressions. It's considered that を is omitted in these cases.)