Differences between moving build plate vs. extruder
Without going into too much detail, since this is a very exhaustive topic, I'll write some pro's of each down from the top of my head:

Cartesian XZ hotend, Y bed (eg. Prusa Mendel):


easy to build (relatively)
easy to maintain
easy to modify
understandable kinematics
with the right frame, no x-y-z orthogonality (90 degree angles) needs to be adjusted
affordable
bad for timelapse recordings
print quality will theoretically always be inferior at the same speeds and accelerations to kinematics that have less mass to move (heavy printbeds will lead to ghosting)
z-wobble is only existent in this approach
big build-plates are no option for this design (last feasible size might be 20x30 cm)


Cartesian XY hotend, Z bed (core-XY, sparkcube, Ultimaker, Makerbot)


less mass to be moved -> faster print speeds possible
almost no size limitaions
construction is easy to enclose in most models due to the cubic frame
looks almost always professional
enclosure can be hard to modify due to constraints in space


XYZ hotend (Delta bots)


master of circles
less mass to be moved -> faster print speeds possible
impressive to watch
more load on the processing unit due to more complicated kinematics (32 bit needed for fast print speeds and responsive control with display)
kinematics not easily understandable
error-cause search can be very complicated
more accurate in the center than on the outer limits due to the kinematic approach


The list is for sure not complete, and as a major disclaimer: print quality will always, with every approach, depend more on the setup and calibration of the printer than on the model. There are people around that produce great prints from an acrylic frame cartesian printer and lots of people that produce mediocre results with expensive printers in fancy designs.

I will add some links to the list items when I find the time, for now you have to believe me. I am highly appreciating corrections and additions!