It allows for a 4 valve arrangement with less reciprocating mass. Though the downsides include a taller engine (or wider in the case of V engines). Pushrods CAN be used to get a 4 valve/cylinder arrangement, an example of this setup is the 24V Cummins turbo diesel that's in the Ram, though it makes for alot of reciprocating mass can can't rev as high.
IMHO, DOHC in most passenger cars, especially large cars, trucks, and SUVs, is largely marketing hype.
An example of this:
GM had a DOHC 3.4L V6, the pushrod 3.8 was smaller, about the same power, more torque, cheaper, and smoother. When it came time to release the Olds Intrigue they put the 3.8 in. The lack of the "DOHC" label hurt sales.
Next time you're in a car w/ a tachometer, watch how far it goes up with normal driving, with high reving cars you rarely go over 5000RPM, with torquey cars you rarely go over 4000RPM. DOHC's main benefit is over 5000RPM.
It's not bad tech, just overrated.