What are the advantages of towbarless pushback tugs?
Advantages of towbarless pushback.


You don't have to store different tow-bars for different a/c types. This is especially good for ground handlers who handle different airlines with different aircraft types. Also handy for ad-hoc one-off flights.
Removing need for tow-bars means one thing less to check daily, weekly, monthly, annually.
Towing with a tow-bar requires a man in the cockpit to ride the brakes (brake-rider). This also means the aircraft may require power (either from APU or from the towing vehicle) to supply hydraulic pressure. An aircraft towed with a towbar is dependent on the aircraft brakes if a sudden stop is required as the towbar will not be able to take the weight of the aircraft on its own on decelerations.
A normal tug (which uses towbar) needs to be very heavy to enable it to tow an aircraft. Without the weight, the tug will just spin its wheels. The tow--barless tug will have the weight of the aircraft on its wheels so it doesn't need to be as heavy. Just solidly built to support the engine and associated hydraulic stuff. This weight also makes the brakes more effective.
A towbarless can tow the aircraft faster as control is way better. In some busy airports they only allow you to move the aircraft across busy taxiways or during peak hours if it is either taxied under its own power or towed by a towbarless tug. Towing using tow-bar is not allowed as it is very slow. 
The tow-bar has 'shear pins' which limit the force which can be applied through the nosewheel towbar points. This reduces the towing speed. They can and do break once in a while. While a spare is usually carried in somewhere on the 'bar it still stakes a few minutes to get in installed. Not fun in the middle of a pushback and may result in a blocked taxiway.
A lot of injuries are caused by towbars. Either they jacknife when something fails or a lapse in procedure caused the nosewheel steering to be activated before the bar has been disconnected. The danger here is the nosewheel is self-centering in almost all aircraft and will swing the bar if the nosewheel is not centered. Scary to see and can take your legs out.


I'm sure there are more aspects but these are the major points. Hope this clarifies a bit.