The Wii U Pro fills the same role as the Classic Controller Pro from last gen- an alternative pad for those uncomfortable with Nintendo's default controller- but the difference is that this time the default Wii U controller is itself a fully functional dual analog gamepad. Because of this, there are only a few reasons to purchase the Wii U Pro:
1. You intend to play multiplayer games and don't want to use/don't have Wii Remotes for 4 other players.
2. You're not comfortable using the gamepad or the touchscreen for whatever reason.
3. You're interested in using the Pro as a PC controller.
Otherwise, there isn't much of a point; the components in the Wii U gamepad and the Wii U Pro are mostly the same with few differences.
The grips are much fuller than those on any past Nintendo controller (including the gamepad) and have similar contours to the Xbox 360 controller. The analog sticks are the same new clickable type used in the gamepad, which resemble Nintendo's Nunchuk/CC Pro sticks but have much more precision and smoother movement; they're the best sticks Nintendo has ever used by a wide margin. The D pad is a little more stiff than the gamepad's, but it performs just as well and is far, far better than the D pads in the 360 or even the Dual Shock 3. The Z shoulder buttons are much more narrow and trigger-like than the gamepad, and L and R are 360-style bumpers. The face buttons are the same as the gamepad's, with a softer "click" and tighter grouping than those on the CC Pro (I consider these changes improvements).
Many people dislike the placement of the buttons below the right stick, but this layout makes more sense for modern gaming than a staggered layout; modern games make heavy use of the right stick for aiming and camera control and use the triggers to perform most basic tasks, leaving the face buttons used for less frequent tasks like ducking or reloading. Giving the right stick a higher priority makes sense considering that it's used more often than the XYBA buttons, even if it is awkward to adjust to the change.
Unlike the CC Pro, the Wii U Pro is fully wireless and has its own rechargeable battery. Nintendo's claim of 80 hours of life seems accurate- I recently cleared Resident Evil 6, Arkham Origins, and Mario 3D World on a single charge over the course of several weeks (with rumble on) and the battery still isn't down to the lowest bar yet. Unlike Microsoft and Sony's current pads, the WU Pro only has one rumble motor, housed in the right hand grip, but the rumble is evenly dispersed and the motor is much more quiet than the one in the gamepad. The rumble is smoother and heavier than the gamepad's, although not as responsive for very brief rumbles.
There are a few small, subjective problems with the layout of the Wii U Pro. The smaller and more narrow design of the ZL and ZR buttons seems like an overt attempt to copy the triggers of the Xbox 360 controller, the issue with this being that the ZL/ZR triggers on the Pro are not analog. Using digital triggers on a hinge feels somewhat awkward, as they don't protrude as far as analogs and must be depressed at their very bottoms to get a firm click. This is not a large problem, but I find myself wishing ZL and ZR were flat buttons (as on the CCPro) rather than hinged faux-360 triggers.
Another problem is that the XYBA buttons are placed further inward than they are on the gamepad, which, combined with the angle of the grips, makes reaching the innermost face buttons more of a stretch than it should be. Virtual Console titles that require players to hold the Y button to run and B to jump may feel awkward as a result, to the point where I've remapped these controls to X and A for several games; in general, X and A now correspond to the positions Y and B used to occupy relative to the right analog stick, which may end up causing confusion over whether or not to reverse the X/Y and A/B buttons in certain games.
Despite a few minor complaints, the Wii U Pro is a very solid, comfortable controller that can even hold its own against the more fully featured Xbox One and Dual Shock 4 controllers. It's usefulness is very limited for players who are satisfied with the gamepad, but it can be a great alternative for those who want something more conventional for single or multiplayer games and it can even serve as a superior substitute for the 360 controller when connected to a PC (which requires an adapter for most games).