I am answering this anonymously, also for obvious reasons. As a referee, I get quite annoyed when, after having recommended rejection for a paper, and given several detailed reasons for the rejection, the same paper comes back to me from a different journal with no changes at all. In other words, the authors didn't even try to address my objections. In this case, I have absolutely no qualms about rejecting it all over again. On the other hand, if the authors genuinely tried to answer your objections, and you still think it is not acceptable in the new journal, maybe you should ask the editor to send it to a different referee.
It is in principle possible that reviews are revised/replaced (as I assume you know).
However, as far as I know, this is mainly done to replace factual errors. From your description it seems to me that your situation might be at the borderline of factual errors and different interpretation.
You could also check http://www.ams.org/mresubs/guide-reviewers.html (in particular the point 'evaluative reviews' to see whether the review is in line with what is written there).
Yet, since you say that the review is not negative, this seems less relevant to your situation.
Now, in case you decide to do anything, in your situation it seems definitely advisible to contact the reviewer directly. As it was not negative it might be a misunderstanding/an honest mistake, and the reviewer might be willing or in some sense even happy to correct the review if made aware of the problem.
I am a fairly frequent reviewer for MathSciNet. My greatest worry doing this is to write something wrong in a review, and by doing so, somehow make a fool out of myself. So, in case an author of a paper I reviewed should ever contact me in a friendly way and point out why my review is not to the point and I then understand that s/he is right, I certainly would try to get this corrected, and be grateful for being made aware of it.
By contrast, I know indirectly (a collegue of a collegue...you get the idea) that if the reviewer does not want to change the review it can be very difficult to get it changed (except there is a direct factual error).
So, you need the reviewers consent anyway, and as André Henriques says, there is no way to hide that you were 'behind' the activity; or even if there was a way to do this anonymously, it should be the default assumption that the author of the paper is the one 'complaining'. This seems another reason for directly contacting the reviewer, as it seems rather more likely to get her/his consent by this course of action, then by one going an official way.
How to phrase the communication with the reviewer: a way could be to not directly or mainly critizise the review, but to just sent the reviewer your point of view. In other words, write somehow your own review and share it with the reviewer. Perhaps only pointing out key differences in passing.
In any case, if you do not care deeply I would at most contact the author of the review; perhaps this works out well, but if not rather drop the matter as further activity might be in vain, and not be worth the effort and negative side-effects.
Best Answer
I have done this as a reviewer, and have had it done to me as an author. I have no problem with it. I do think you should disclose to the editor that you've reviewed the paper before. (This can create the awkward situation of having to say "I reviewed this for journal A, and didn't think the theorem was interesting enough, but it fits just fine in your journal." But I found the editor in question to be not at all offended.)