Is there a semantic difference between "manipulable" and "manipulatable"?

You neglect that "manipulate" already has a suffix. The word "manipulatable" applies a suffix to an already-suffixed root.

Consider the difference between the known meaning of the word "manipulate" and the supposed meaning of its root. "Manipulate" describes a process of controlling something. That process part is very important--it's what the affix "-ate" brings. The root, then, means at its essence the inherent property of being controllable. We don't have a word that's based on this root for that, but that's important to understand for appreciating the difference here.

"Manipulatable", therefore, means that a thing can be worked especially because the manipulate root is already affixed to describe the act of controlling something. When you say "manipulatable", you're describing something that can have that process applied to it. Thus the relevance to mechanical processes. This definition is used much less often than "manipulable" is is likely far newer; the distinction is likely maintained overtly and intentionally, to provide a more specific word than "manipulable" brings. Not all sources recognize "manipulatable" as a word.

The word "manipulable", on the other hand, skips the "-ate" affix and goes a step backwards to its root. It forgoes description of a process or action. Instead, it's describing something as "capable of being controlled", which makes sense in many more contexts besides those that can be described as "manipulatable".

So there you go. All things that are "manipulatable" are also "manipulable", but not all things that are "manipulable" are "manipulatable".

Based on a comparison in Google Ngram Viewer, 'manipulable' is far and away the more commonly used word.

Sometime a discrepancy like this can indicate that one term has a more arcane or specialized meaning, as you suggest. A cursory browse through the source material provided by the above link, however, doesn't show me anything clear cut. It looks plausible that 'manipulatable' was a common term in mid-twentieth century behavior psychology, but there are also similar texts in the 'manipulable' sources.

My conclusion would be that 'manipulable' is just the older and more common word. The appropriate context for one word over the other is unclear, but the meanings are essentially equivalent.