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".
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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:
I hope this helps!
I think "maneru" is a more formal written style and "mane suru" is the conversational version.