JabRef has support for updating arXiv, MathSciNet and a bunch of other identifiers (as well as revising existing bibliographic metadata with updates from web services).
There are two options.
First, select the entries and then use "Tools > Search document identifiers online > arXiv". This will automatically query the arXiv and puts the identifier in the eprint field.
Second, in the entry editor, choose "Update with bibliographic information from the web" (the circled arrow on the left toolbar in the entry editor). This will query the selected database and updates all bibliographic metadata. Usually, I use this feature to see if a preprint in my library is already published but it also works the other way around to discover the arXiv identifier for a published article.
Disclaimer: I'm one of the core developers of JabRef.
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
Google scholar is free; MathSciNet requires a subscription. In practice the usual effect of this is that one needs to access MathSciNet via a university IP address rather than one's home address. But it also means that one can't provide web pointers to MathSciNet searches or reviews and expect them to be usable by people who are not themselves professional mathematicians; it is possible to link to bibliographic entries for individual articles but non-subscribers are not shown the review text, only the bibliographic data.
MathSciNet covers essentially all mathematics journals; Google scholar covers only what it can find online. On the other hand, Google scholar covers unpublished preprints and some published mathematical material in related disciplines (e.g. theoretical computer science conferences) that is not as comprehensively reviewed in MathSciNet.
MathSciNet is indexed only by title and abstract/review text; Google scholar is indexed by the full text of the article.
Some articles in MathSciNet have a review written by a knowledgeable reviewer, that puts the article into context better than the authors did. (On the other hand, many MathSciNet entries merely repeat the authors' abstract.)
MathSciNet has much more reliable publication data than Google scholar: its BibTeX is generally usable as-is, it properly collects papers by the same author and distinguishes papers by different authors, and it doesn't have duplicate entries for the same paper. However, Google scholar generally has better citation data than MathSciNet: although MathSciNet lists the papers that cite a given paper, the ones that it lists are generally a small subset of the ones Google scholar finds.
Google scholar will provide links to as many different online copies of a paper as it can find (e.g. preprints from the author's home page); MathSciNet will only provide one link, to the official published copy, and will do so only for a subset of the journals it covers.