Is there any difference between 'P' mode with appropriate settings (like aperture etc.) and specific scene mode?

Yes. Speaking from a Canon perspective, P is a "creative" mode and therefore offers a number of settings that aren't available in the "scene" modes. In P you can select your own aperture or shutter speed and the camera will use information from the meter to adjust the other settings to compensate and provide a reasonable exposure. In scene modes like "sports" or "portrait" the camera is fully automatic and will choose those settings on its own, with the particular choices optimized for the type of scene you've chosen.

Many of the other settings, like white balance, ISO, metering mode, and autofocus setting are also out of reach in the scene modes (and other fully automatic modes, like The Green Box). These are all available in the creative modes, including P, Av, Tv, and M.

Other makes and even specific models may have their own takes on the specific differences between modes, but Program mode (that's what P is for) generally lets the photographer adjust shutter speed or aperture.

Now, it's entirely possible that you might get similar results from P as you do from the scene modes if you happen to make choices similar to what the camera chooses in the scene mode in question, but P gives you the creative control to make your own choices while still getting a good exposure. It's basically a marriage of aperture priority and shutter priority modes, where you can choose to adjust either aperture or shutter speed on the fly and the camera compensates with the other. One difference from Av or Tv is (as Michael Clark points out in his comment) that in P mode, the camera will select starting values for both parameters.

So is it always better to use a scene mode then 'P' mode for medium skilled photographer?

No. Many people would say the opposite, i.e. that it's generally better to shoot in P mode than in the scene modes because you have much more control. The scene modes are there to make it possible for even the newest camera owners to get decent shots, and if you find they do what you want and you like using them, you should. But I think most experienced photographers would agree that the shots they get are more likely to match their expectations when they're in control.