Yes, they have different meanings. In the context of Fifty Shades, I don't do romance means the character is wholly uninterested in the usual trappings of attracting a mate: wining and dining, emotional intimacy, evoking tender emotions, etc. The speaker is solely interested in sexual activity. On the other hand, someone who says I'm not romantic might be interested those things but not very skilled at them.
I don't do X means I'm unwilling to engage in X. It connotes unwillingness and does not generally convey inability. See the extremely topical question "What does 'I don't do dogs' mean?" for an in-depth exploration of this phrase. This is a modern construction, so it's unsurprising that you didn't find any hits in your N-gram search.
I'm not romantic expresses a more fundamental statement about the person's nature. Romance is alien to the speaker's personality and emotions. It would be difficult, and perhaps impossible, for them to experience or express it.
There's also a subtle difference between romance and romantic. In this case, romance is an activity, while romantic is a descriptive quality. An unromantic person might well be involved in some romancing, but it would likely either be awkward for them to attempt (because romance is not part of their nature), or one sided (someone else attempts to romance them, but encounters difficulty, because they don't experience romantic emotions much or at all).
Conversely, someone who doesn't do romance may be fully capable of experiencing romantic feelings, but they would surely stifle and not express them. Likewise, they would categorically reject all attempts to romance them, whereas someone simply unromantic might warm to the idea, or at least appreciate it.