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!