They both have advantages and disadvantages (electric or diesel)
Push-Pull
Advantages
The loco can be swapped out when it needs maintenance, at a different cycle to the passenger stock
Individual faulty cars can be removed or swapped out without taking the whole set out of service
Changes in length of the consist to suit changing requirements is easier
For diesel, no underfloor prime mover vibrations or noise for passengers
Disadvantages
Locomotive takes up platform room when consists get long and space is limited (not sure how much of an issue this is in the US, it is in the UK)
Although the total weight of a consist is probably much the same, the weight of the loco is concentrated in on place, thus causing more track wear (this pushes the cost of operating Push-Pull in the UK, as the track access charges for a loco+unpowered cars is far higher than for the equivalent number of MU cars)
Loco is a single point of failure, if that dies, the train can't move
MU
Advantages
All vehicles can carry passenger, no 'dead' train length
Less track wear due to better spread of weight
Distributed traction means better resiliency, if one fails the unit can normally limp home using the others (more so with diesel than electric)
Can allow multiple unit working with end doors and diaphragms fitted
Diesels can save fuel by shutting down one or more engines when they are not needed.
Disadvantages
If one vehicle is faulty, the whole set is out of action
Diesel MUs provide a less comfortable passenger experience
Impossible to gradually extend or reduce train lengths to suit loadings short term, and harder long term too.
Performance is mainly down to the amount of power installed, however in very high power/speed cases, an MU will have more powered wheels to get adhesion
I'm sure other people can think of some others.
I've been commuting on DMUs of various types for over 10 years, and I still can't ignore the engine noise and vibration.
http://en.wikipedia.org/wiki/British_Rail_Class_220" onclick="window.open(this.href);return false;
http://en.wikipedia.org/wiki/British_Rail_Class_221" onclick="window.open(this.href);return false;
http://en.wikipedia.org/wiki/British_Rail_Class_222" onclick="window.open(this.href);return false;
http://en.wikipedia.org/wiki/British_Rail_Class_158" onclick="window.open(this.href);return false;