If I want to start a new website, the first thing I do is check out the competition to see both what they are doing and how much traffic they are generating. I would normally use a site like google.com to find the competition, and then a site like compete.com to find their traffic stats.
Now if I want to develop a new iphone app, I can use the cruddy app store search and some third party sites to find the competition, but how do I find out how many people are using those apps? It seems the only way to judge popularity is through the number of ratings. Are there sites like compete and quantcast that can give me statistics about the number of unique users an app has each month or at least how many downloads?
I'd suggest using the reviews as a way to extropalate interest. If there is one that publish the numbers of sales take the reviews as the denomoinator and the sales as the numerator to come up with a multiplier. Use that multiplier against the quantity of reviews for other areas and you have a swag mechanism that is better then nothing.
Also if you don't find anything then I'd go to another sight that offers both # downloads to # of reviews and come up with something that way say like download.com. This gives you a number but it will be highly suspect because the review rate to downloads is so low, but it gives you a tool rather then no tool at all.
I think that Apple is keeping this information strictly confidential for a lot of reasons. There seems no way to extract such information unless the creators of the App publish it.
When doing market research I recommend using AppCod.es (shameless plug ;) - it's faster than typing in the query in phone, allows you to track your app position in the store later on, and also to keep a list of competitors in one place.
I also use Google AdWord keyword tool, to estimate how much demand is there for a given search phrase.