What are the benefits of a PCB over other methods?
PCBs typically have lower parasitic inductance and capacitance than a breadboard.  

Impedance can be controlled for high frequency signals.  Circuit density can usually be higher.  

In circuit and functional testing is easier.  Manufacturing is WAY easier.  

Conducted and radiated EMI will be less for a properly laid out board.  

Reliability of a PCB should be better, especially if there's shock and/or vibration.  

There are other benefits that I'm not thinking of at the moment, but I'm sure others will jump in.

Of course it's possible to make a complete mess of a PCB with poor layout and manufacturability.  PCB layout is a skill that requires a lot of experience for all but the simplest least critical circuits.