I'm just wondering if they are the same word or if there is an actual difference (in meaning or nuance). It seems weird to me that there would be two different verbs with the same kanji stem that mean "to imitate".
They're not the same word, but it's not a coincidence they're so similar. They share the same origin (真に似せる) and have only diverged slightly since then. One word is the verb まねる; the other is the noun まね plus the verb する, turning it into the suru-verb まね（を）する.
まねる means to imitate or copy. In most of the dictionaries I checked, the definitions for まね began by referring to まねる. That is to say, まねする can mean the same thing as まねる. However, まねする appears to have two meanings まねる does not:
To act or behave. From the examples in the Kenkyusha dictionary, this seems to have a distinctly negative connotation. I've picked out a couple examples:
- 「ばかな真似をする」 act foolishly [like a fool]; play [act] the fool
- 「乱暴な真似をする」 behave rudely; use [resort to] violence
To pretend or feign. I'll pick out a couple examples from the Kenkyusha to illustrate this, too:
- 「驚いた真似をしてみせる」 pretend to be surprised
- 「おびえた真似をして『キャーッ』という」 scream in mock fright
I hope this helps!