A piano is a keyboard, technically speaking. Keyboards includes everything that has black and white keys - the digital ones and the acoustic ones (pianos, harpsichords, organs). Both can be used to play "chords" and "running notes/ arpeggios" - it is a matter of the genre and type of music one is playing. Many keyboardists use weighted-keyed digital pianos which is very similar to the acoustic piano in feel, thereby narrowing the difference between playing an actual piano and a digital one. In fact, most pianists are very particular about the feel of the keyboard and will like to have one that feels as close as to the real thing as possible.
Why digital? Portability and the ability to play other sounds other than piano. It is not true that digital keyboards responds slower than the acoustic ones. In fact, the unweighted keyed keyboards responds much faster because of the lighter keys. For example, playing repeated demisemiquavers on one note is pretty difficult on a real piano (not impossible) but relatively easy on an unweighted keyboard (this technique is commonly used when playing the B3 organ).
For either digital or acoustic pianos, faults and errors can be picked up easily by the listener if they lack the skills to play a technically difficult piece. Of course, if all they play are pads and sustain chords, the possilibility of making errors are slimmer.
then you'll have the most realistic piano sounds you'll ever have.. 
