What are the reasons why the mobile apps market is more developed and active than the desktop one?
It's not. There remain vastly more desktop users than mobile users;
Now those numbers are from last December and Mobile is gaining fast, but it's too early to say Mobile "beats" desktop. It's also a facile argument; mobile's gains have been so fast because they're a new market sector. Apple's OSX never really took major ground away from windows because it's fighting an established competitor; most people really have to choose between Windows or OSX. That's not the case with mobile because it's new. Windows, OSX and Linux users alike got iPhones.
Now, the "more developed" argument is absurd. Desktop operating systems have been around for over thirty years. Sure, Mobile started off more polished and usable than the good old Commodore 64.
What you're seeing here is a matter of visibility, not activity. Mobile is visible. Everyone is talking about it. It grows in leaps and bounds because it's popular and new. It's emerging technology. It also iterates fast; people buy new phones every year or two, maybe three. People keep their desktop PC for years, many only upgrade their desktop when it breaks.
Now for apps? How are you counting how many desktop apps there are? Because there are a lot of them, and there's no central "store" for them. Not even on OSX which has recently moved toward that; there's still tons of software out there on thousands of random sites. We'll likely never know exactly how much desktop software there is out there.
But app stores? The count is easy as pie because it's a walled garden. All you have to do is get official numbers from Apple/Google/Microsoft. There just aren't official numbers for desktop applications so it's simply not fair to compare a well known figure for mobile to "I don't really know" for desktop.
Mobile is in fact developing so fast because it's such an immature platform. The first iPhone didn't even have Copy and Paste! It was briefly treated like that wasn't even important! Mobile was so new that no one, not even Apple, despite lots of research, really knew everything that was important. Everyone in mobile is still learning. It's not developed, as evidenced by Android's complete redesign last year. Mobile is no where close to "developed". Apps aren't either; a bunch of companies, even Facebook, are looking at their mobile apps and wondering "how the hell do I make money out of this?" Ad monetization on mobile's still a wildcard at best.
Now, mobile is learning fast because the focus is on the users. Desktop rarely had all that much focus on the users until fairly recently. User Experience was hardly considered in the 80s or even 90s, and it certainly wasn't the hot new thing.
Even with the latest advancements, mobile websites remain limited and on a native app you can accomplish things unavailable on mobile web. This is not as true for a regular browser vs a desktop app. There are still limitations even to a regular browser, but they usually have to do with computing power or with advanced interactions, which aren't that common among consumer-facing applications.
A specific case of this is that one of the reasons to develop a mobile app has to do with the unique functionality of mobile devices - such as the camera and location-based features. A desktop app can also provide additional value, such as direct access to scanner/printer or other peripherals, but they aren't in such big demand either.
BTW, your claim is correct regarding the consumer market, but if you examine the B2B market, you'll see that desktop apps are much more common than mobile apps, by several orders of magnitude.
There is a number of reasons, some of them are technological (as described by Vitaly Mijritsky) and others are more human and business related.
One of the most important is that mobile apps are still trendy and there are a lot of white spots and ugly applications in certain market niches, lot of users track new apps and install them to find the best one. So the developers and their mobile products have more chances to get noticed (and making online mobile app is simpler/faster than making real mobile app).
Also the world gets more mobile in common, so add to first reason multiple cases, where existing applications (nor desktop or web) is possible to use "on the go", so instead of adding new tasks and functionalities to rather advanced desktop/web application, attention of developers is payed to transforming current app usage to mobile needs. This is typical for business apps — for instance, last desktop version of Apple iWork office suite was produced at 2009, and mobile versions (both for phones and tablets) are actively developing nowadays.
So mobile market is not so filled by apps as more old and developed web and desktop markets, and when the difference among them will get less strong, we will see less difference in amount of mobile apps (both mobile online and native mobile) and desktop apps — until another newtechnology (brain connectors? :) will get to mass markets.