Very good question. I'll try to give the complete story here. I'm a developer myself so I might get a little technical now and then, but only that will show you the true reasons behind all of this.
Baseline: There is no distinction between "phone apps" and "tablet apps" in the Market.
A bit of history first: Android was meant for phones (let's say a G1). It was a scope thing. Then it started to support different screen sizes and different screen densities for these phones (like the Nexus One, at the time of Android 1.6 Donut). If you want to know why I'm not using "screen resolution", then read this. Finally there was a market for other devices, like TV's and tablets with different needs and extra large screens. TV's aren't different from the phones in that they just show the same screen image, only enlarged since you sit further. Tablets however almost require a different form of interaction and have extra large screens combined WITH high screen densities, as do computer screens. True support for these tablets only came with the last version currently available, Android 3.0 Honeycomb. The only well-known exception being the Samsung Galaxy Tab which uses a different trick, just like the TV's actually, which I won't go into right now.
Transparency: The idea is that every app should be able to run on any Android device, if only the developer stays within reasonable boundaries, as defined by the platform API. This really is obvious for developers, and hence there is no good reason to cross the line here, but some still do, which makes for a small percentage of apps working really badly in general, or working really badly in any configuration but the specific one they were built for. This is a shame because these different configurations are really commonplace nowadays. Then again there are criminals in every society so let's keep these out. Next, there are several levels of support for tablets, you could say:
Apps that were built but never adapted for any of the above story. Let's say these are version 1.5 apps that haven't been touched since. For this Android uses a compatibility mode. If the screen density is different than what the app was intended for, and the default used to be medium, than the app is being rendered scaled, either smaller or bigger. The whole picture here is being enlarged or shrunk basically. Tablets have the same medium density so they won't show any of this. If the screen size is different, and it is extra large for tablets, than a black border, like what you know from your widescreen TV, is being added. Would the screen be smaller than intended, than the app is filtered out via the market for that device, because there simply is no compatible way of properly displaying this app and have it work well enough. Plenty of caveats, but it works well. Still, you could call these app developers pretty bad, because the effort to at least get a typical app out of this compatibility mode is fairly small, as long as its developer used native widgets and didn't do too much fancy stuff. You can compare this btw to the way iPhone apps are displayed enlarged (switchable) on an iPad, when they are not adapted.
A step up are apps that have information incorporated about which screen parameters they support. As the first of this was introduced in 1.6, this is still very much in the context of phones, but anyway still applies to tablets as well. What these apps will do, or rather, let the platform do for them considering they used the proper advised techniques, is scale their content. Now I'm not talking about scaling up the image in pixels. What it means is that if the app used to display 5 items on a regular screen, they will now display 10 or more items in the same view on a larger screen. As an added effect, all the controls, such as buttons, and the font size, stay more or less the same between all these devices. This is very much intentionally since we are using finger touch to operate them. If a button would become smaller, your finger might be to big for it; it would be awkward. The perk for tablets is that they display more information. I personally already call this very much tablet compatible, and if so, there are way more than 16 or 100 apps working on tablets, if not the bigger part of all the available apps. Don't let the marketing fool you. Do these apps make use of some of the new ways of interaction that comes with tablets? No they don't. So are they "designed for tablets"? Eh well, ..., it depends on the app. Many show up near-perfect regardless. Some devs include specific artwork for all the configurations (this is advised) so that everything looks nice and sharp. Some devs even test on all configurations :-) (which is also recommended). But the next category of apps really is designed for tablets:
These are apps designed specifically also with tablets in mind. They make use of some newer API parts that allow them to better fit in and to allow them to display multiple sets of information next to eachother. Android/Honeycomb doesn't have windows that you can drag around, but instead has parts of a screen that display each their own stuff, called "Fragments". Note that this level of abstraction is below what Android used to have, a single "Activity", whereas on a desktop computer, the "window" concept is sort-of supra-application, an added level on top of what we used to have on old DOS computers. The need to display even more information than what I mentioned in paragraph 2, comes from the fact that there are a lot of one-dimensional lists in Android apps of course. In 2 these would extend by adding new items in the length, but in width they would just fill the screen. Add to that that tablets are preferably held in landscape mode, and these lists with very wide items would just be unfortunate. Hence the idea to display multiple lists next to eachother, and make it possible to slide them in and out, etc. I.e. make use of the free extra screen space. The same rationale is behind controls moving into the screen space, the menu being integrated into the top bar, and notifications getting more screen space in the bottom bar, all things these apps can support on top of what they supported on phones. There are however forcibly no apps that ONLY work on a tablet. There is also no per-device filtering in the Market currently, only a filtering based on these rough specs and parameters we have been discussing. The recommendation is to add this support on top of supporting phones, and it is perfectly possible to do this in the same app. There is however a somewhat exceptional situation in that Android 3.0 Honeycomb is currently not released for phones, only for tablets. But it has been confirmed that in the future these two forks will likely merge again, and the reason for the fork has been an organizational one rather than a conceptual one, where one group focused on getting tablets, whereas another kept working to improve Android for phones to bridge this long time period. Though I have to admit, personally, I see this step up to full tablet support as the biggest change in Android yet. No way you could say this doesn't take any effort, or doesn't induce some mild headaches, such as maintaining backwards compatibility. Can't blame Android, it is exactly what it only could have been, but it is non-trivial nonetheless. So please bear with us, users! Things are always very much in motion.
So, to get back to the original question. How to distinguish these in the Market? In short: you can't. The Market has advanced filtering capabilities, but since nearly all apps can run in some compatibility mode or another they will show up on your tablet. The mindset behind the Android Market, from Google's point of view, as well as the OHA, has always been to interfere as little as possible and have it open. Since the intent is to build a community and ecosystem, I guess they meant to rely on news sites a la AndroidGuys to do some filtering for you, by reviewing and testing apps and rating them. That's where you should look for now.
One more thing: Now, why does Apple do have this kind of differentiation? There's a simple reason for that! If you objectively look at iOS devices, then you'll notice there are only a few screen resolutions. I believe 2 between the iPhones, and 1 for the iPads. While the Android platform now truly supports the full scale of screens - you could throw any resolution at it and it'd work perfectly - , iOS can't. That's why the new iPad 2 did not get a slightly bigger resolution. It'd basically trigger a totally new category in the App Store, as well as a rewrite of any app, to make it run in anything but compatibility mode. Guess I don't need to explain how much that sucks from both a user and developer perspective?
To summarize: as usual, the Android story, as is probably the real situation, is a bit more complex, but you get served well in the end.
Full reference: http://developer.android.com/guide/practices/screens_support.html
Expiration date: 2012-03-07