In simplest terms, MIDI is the message system used by the equipment. USB is the physical communication link over which those messages are sent. Drawing a parallel to computers and the Internet, MIDI would be like TCP/IP, while USB would be like the Cat5 Ethernet.
The P105 is "MIDI-compliant", in that it will transmit and receive standard MIDI messages; what it lacks is the older DIN 5-pin connectors traditionally used to connect MIDI-compliant hardware (as well as a host of other things, from computer keyboards to printers, in bygone days). Instead, the MIDI signals are sent to the computer over the USB connection, with the OS creating a virtual pair of MIDI input/output ports for MIDI-compliant software to use to interface with the device.
MIDI-over-USB suits the needs of the average home recordists, who use the keyboard to control computer software (such as a DAW, softsynth or sampler). The USB connection, however, doesn't work with most hardware MIDI modules, such as hardware synths or step sequencers. You almost certainly don't have any of these, but it can be a concern to vintage enthusiasts who score a rackmount analog synth and then find their keyboard can't control it (or the round trip through the computer introduces unacceptable latency).
So, your P105 and Launchpad can be used together, and the Launchpad can even be used to send messages to the P105 (and vice-versa if you really want to), but you'll need a computer (with the appropriate number of USB 2.0 or 3.0 ports) and probably some software (the video's using a DAW application, Ableton 8, to handle the looping and layering). You can't just plug the Launchpad into the P105 and expect them to talk to each other.
As for product recommendations, this isn't really the place. I will say that the P105 is very good at what it is, which is a home-quality digital piano, perfect for student use. It's about as inexpensive as you want to mess with in that vein; there are cheaper, in all senses of the word, but you'll be able to tell. It isn't really designed to be used with a DAW, although it has the basic ability to interface with a computer for recording/playback of MIDI signals. If you want to save a little money and don't mind needing a computer to make noise, look for a good used hammer-weighted MIDI controller in the $400-$600 range; the money you're spending there will be in the quality and feel of the keys, instead of into the patch library (which you probably won't be using with the keyboard plugged into a PC), and you'll probably get a few additional controls for DAW/track control that you'll no longer have to map to the Launchpad.