MathSciNet vs Google Scholar What are the pros and cons of the MathSciNet database vs Google Scholar?
I don't have access to Mathscinet so this question is out of curiosity, and also this question where MathSciNet is used to find paper counts. I reckon Google Scholar will almost always be the more comprehensive of the two with higher paper counts for any particular author and includes papers that don't use the MSC codes as well papers from other subjects that may be of mathematical interest but aren't included in mathscinet.
One definitely annoying thing about Google is that in the advanced search it doesn't have a mathematics only category, but lumps it in with computer science and engineering so you sometimes need to add something like -"computer science" -"engineering" mathematics to your search term to filter out unwanted results, which isn't ideal.
 A: *

*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 On the other hand, Google scholar covers unpublished preprints and some published mathematical material in related disciplines (e.g. theoretical computer science conferences) that is not as comprehensively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.
A: I find it a mayor drawback of Google Scholar that it also finds things which are not publications at all (e.g. sometimes it finds talks which end with a bibliography page).
Another drawback of GS is that it gets confused by some preprint-series I know, namely the ones which attach the list of all the preprints in that series. In these cases GS counts these as citations which is just wrong.
However, I find GS helpful and use it frequently but I would not consider its output reliable.
(Another related issue is, that GS sometimes identifies weired things as the author, e.g. try to find papers with "Mathematik" as author...)
By the way: Wouldn't it be a great feature of both GS and MathSciNet to give snippets of the places in which a paper is cited? Imagine, if you click on the "cited by"-button and see a list of paragraphs or sentences is which the citation happens...
A: To add to what is already mentioned, if you are not searching a specific paper but want to search by a few approximate keywords (or an approximate author name) for instance, Google Scholar seems to be better at finding exactly what you had in mind, while MathSciNet is somewhat rigid in the searches.
A: One disadvantage of Google Scholar is that it is not quite reliable in recognizing the same paper coming from different sources (arXiv, multiple preprint servers, journal website, the author's homepage, etc). This leads to a citation overcounting and unreliable metrics.
On the other hand, the author can update his own profile and merge all these duplicates. In this case, Google Scholar is quite accurate and powerful. Since not a lot of authors do this, unless Google implements some automatic merging algorithm, the metrics it provides are not very reliable (for what they matter, anyway).
A: I think it is worth pointing out that one can acess the reviews at Zentralblatt MATH without subscription. It doesn't show as many results if you don't have a subscription, but if you know  exactly what review you are looking for, this is usually not that big a problem.  
A: MathSciNet is the online version of what used to be the Mathematical Reviews.  It contains not just the bibliographic metadata of the paper (in a variety of useful formats) and a link to the actual article (if it exists), but also a review of the article.   The reviews are perhaps the single most important feature of MathSciNet.
Google Scholar contains the link to the actual article (if it exists) and the metadata can perhaps be extracted automatically -- for example, Papers (a Mac OS X application) does just that -- but it does not contain the review.  Apart from the lack of the review, the main disadvantage of Google Scholar, which probably will become less of an issue asymptotically, is that many entries contain errors in the metadata.
Personally I always check MathSciNet first and Google Scholar second, if at all.
Added
As mentioned in the comments to the question, MathSciNet only contains published work, whereas Google Scholar will list anything which has a direct or indirect (i.e., as citation) web presence.  However (at least in my field) it is enough to check the arXiv for (pr)eprints.
A: Not having subscription access to Zentralblatt, I won't try to comment on its merits.   Leaving that aside, 
MathSciNet is certainly the most comprehensive database of published work in mathematics (and to some extent the closely related areas of mathematical physics, etc.).   It's constructed and maintained by a substantial paid staff of professional mathematicians and technicians, with no advertising revenue.   Though AMS is a nonprofit organization and MathSciNet is partly subsidized from voluntary membership dues, there is no free lunch.  For some time now they've compensated reviewers, who used to work for free, with 8 dollars worth of AMS credits for each review written; it's a token payment but eats into AMS book revenues.
Availability to people outside the North American system of colleges and universities which pay for the service is always problematic; there are frequent arguments in the AMS Council about how to subsidize MathSciNet as well as journals and books for developing countries.    Anyone in doubt about the complexity of AMS finances should try to chat with the incoming AMS president Eric Friedlander, who has long experience as a trustee.
Having said that, I'm one of the lucky people who (so far) have MathSciNet access
and can use my university system password even at home.    Whether our financially stressed library continues the arrangement is an open question, since they have to cancel hundreds of journal subscriptions every year even in stable economic times.    
On the plus side, MathSciNet has developed its search capabilities even though there is usually too little access to the text of articles or books to search that far.    And for a couple of decades the ability to list references in a given paper and citations to the paper in reviews or other papers has grown rapidly.    Another very useful feature is the author profile (even for the elusive et al.), with lists of all published work and links when possible to co-authors, math genealogy.    
Also on the plus side is the professionalism of the staff and the attempt to keep all data precise, even the troublesome identification of authors by sometimes variable or duplicated names (notable for Chinese authors).   
For me Google Scholar is a back-up source of limited use (and limited trustworthiness), which often has the defect of giving too much information in a time-wasting format.    But it's definitely free of charge and will contain even more information if publishers can be persuaded to hand over all content to them
(presumably in exchange for advertising space or priority listing or even old-fashioned cash).       
A: One advantage for me of MathSciNet is that Google Scholar doesn't give me easy access to the full text of an article, even if my university has online access to the full text of the journal, whereas, if I use MathSciNet (I can get access at home via the university library), the library somehow gives me a button linking back to the card catalog entry for the relevant issue of the journal, which will get me the full text in a few clicks. 
Now, this advantage is perhaps peculiar to my university. When I was an undergrad, the library had a proxy, which would prompt you to log in to access something like ScienceDirect, so you'd get prompted for a password clicking on a link from Google Scholar and then have access to the full text. But, as it stands, if someone tells me to go track down a specific article, MathSciNet is probably the easiest way for me to do it.
