What are some reasons for mathscinet listing a article, but then stating that there will be “no review of this item''? In particular does it at all imply that the paper has little or no merit from a mathematical viewpoint?
[Math] When does mathscinet review a paper
mathscinet
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Everyone I know in the AMS would like to make MR/MathSci free, but the problem is that it costs millions of dollars to produce and maintain (it requires a large staff in Ann Arbor and elsewhere, including many mathematicians), and no one has managed to find any other way to pay for it*. This is certainly something the mathematicians in the AMS are aware of and have thought about. The AMS attempts to make it as widely available as possible given the constraint that it has to be paid for. As far as I know, the revenue from MR/MathSci only pays to support it, not any of the AMS's other activities [not so; see below.]
Posters don't seem to realize the huge effort that goes into maintaining a project like this (for example, every article has to be assigned to a reviewer, and every review has to edited). Certainly, I don't believe a free alternative would be able to come anywhere near the quality MathSci maintains, so my answer is no, a free alternative to MR/MathSci is not possible.
Perhaps free supplements to MathSci could be useful, but anything that drew potential reviewers away from MR/MathSci would harm, not help, what is an extremely valuable resource.
*Of course, the intelligent thing would be for the funding agencies in the wealthy countries (US,EU,Japan,...) to pay for MR/MathSci directly, so that it could be distributed freely, but getting them to do this seems to be hopeless.
Added: In response to Anton Petrunin's comment, here are some numbers. The AMS employs 15 mathematical editors (i.e., mathematicians) and a total staff of over 70 at Mathematical Reviews (in Ann Arbor). The total direct cost of producing MR/MathSci in 2008 was 6,569,000USD. However, contrary to what I thought, the AMS does use the revenue from MR/MathSci and its other publications to support a large part of its other activities (Member and professional services, general and administrative expenses). In 2008, about 24% of total publication revenue was used in this way. See 2008 report and ad.
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.
Best Answer
You can see the "official line" here: http://www.ams.org/publications/math-reviews/mr-edit In particular,
In my experience, (and in addition to Thierry's and Andreas's list) this includes:
You ask "In particular does it at all imply that the paper has little or no merit from a mathematical viewpoint?". Well, certainly not, in some sense. A book review, or an elementary survey article, might well contain interesting mathematics, and might well be useful to read (which also one can clearly understand why they wouldn't get a review-- a "review of a review" would be quite silly).
As to my case (2.) above-- yes, perhaps this is Math Reviews (or an editor, or someone who was sent the paper to review) making that judgement. This does seem to be a grey area, as it's not covered by the "Editorial Statement" I linked to (except maybe in the word "elementary"). It would be interesting to get more information about this...