This association of the keyboard with the synthesizer eased its entry into the world of music, but it also placed limitations on how the instrument is played that its designers didn't intend. The limitations of the piano keyboard have been recognized since long before the synthesizer existed. The biggest problem that the keyboard has always had is that, due to the two-row layout with all of the naturals on the bottom row and all of the accidentals on the top row, the performer must usually change fingering in order to transpose a chord from one key to another. This frustrates what should be a simple operation; the guitar player playing a barred chord can transpose it simply by moving up and down the neck, but the keyboard player must keep shifting fingers around to insure that each finger hits on the correct row. The additional manual dexterity and muscle memory requirement makes learning the different keys on the piano a slow and frustrating process. From my own experience, it also introduces the temptation to use teaching shortcuts that cause the student problems later on: a common technique is to start the beginning student out learning the C-major scale, which is played all on the white keys. This introduces a sort of fear or puzzlement at the black keys—what are the for? When does one use them? And then when the teacher starts introducing other scales, the use of the black keys seems arbitrary and unsystematic, and the student gets a bit freaked out. By contrast, guitar pedagogy treats the accidentals as simply other notes in the chromatic scale, which they are, and the guitar student has relatively little trouble understanding how to play different scales and keys.
I'm actually puzzled with (musical) electronic keyboards. Sure, they have a layout like a traditional piano, and yes, the C-major scale is played with all white keys, so why can't if you decide to play, say, a F-major scale, why can't you just remap the frequency of the keys so you can still play it with all white keys? C-major the keys can go C-D-E-F-G-A-B while in F-major, they go F-G-A-B♭-C-D-E and for A♭-major they go A♭-B♭-C-D♭-E♭-F-G. The same fingering, regardless of scale.
Yes, I know you can't do this on a traditional piano, but I'm not talking about a traditional piano here. You can't change the layout of a typewriter, yet it's trivial to change the layout of a computer keyboard (used to type text)—it's a matter of changing the software and boom—you have a Colmak layout!
But short of that, I am facinated by alternative music keyboards, probably because I'm not a musician and to me, these alternative music keyboards seem to show the patterns inherent in music must better than a piano keyboard.