I've been using PostGIS for a few years and only recently started to investigate how I could use MongoDB to deal with certain use-cases. I was dealing with point data that had sparse fields - like OSM data with a varying number of tags per record, and since MongoDB has no schema, it lends itself well to this. I loaded a sample of this data into an instance of each DB and this is what I found.
It appears to me that for simple storage and retrieval of point data Mongo works just fine. The bounding box geospatial queries seem to work well, and I find that the overall performance is very good. It also is very easy to setup and get going, although I have found that the mongoimport tool does not allow me to define a compound 2D coord field in a TSV or CSV file. Since it's pretty easy to write a script that generates JSON, this hasn't been much of a problem. Its major drawback at the moment is that almost nothing else in the geospatial realm can natively read data from it. There appears to be an experimental Mapnik datasource plugin at https://github.com/springmeyer/mapnik-mongo, but that's all I could find.
PostGIS on the other hand takes a bit longer to set up (at least for me), but as was mentioned above, it provides way more features right out of the box. In addition to providing much more sophisticated spatial analytic capability, it is also natively supported by a ton of other applications and libraries; Mapserver, Mapnik, QGis, GDAL, etc, etc. To me, PostGIS is much more a true GIS system, rather than a simple storage and retrieval system.
As far as performance goes, I found that I could retrieve data very quickly from both systems. However, it seemed like PostGIS benefited more from the presence of indexes. MongoDB was slightly faster at returning the entire data set to me (2 million records) at once, and slightly slower at returning a query that used an index - the first time. I'm not exactly sure of the mechanism that it uses for caching, but I can see that if I repeat a query in MongoDB, the results come back much more quickly the 2nd time around. I see something similar in PostGIS, but not to the same degree. I did also note that the memory usage on my machine seems to be far higher with MongoDB running than it is with PostGIS.
So, my conclusion is that I'm not going to get rid of PostGIS as my default geospatial storage and analysis system, but for certain types of projects (namely web maps that display image tiles and/or point data) I may consider using MongoDB as my data store.