Electrically, there's not a huge difference in many common scenarios. If you're working with pure DC, relatively low current levels, fairly low voltages, etc., then you can get by fine with protoboard and soldered wire, etc. The biggest disadvantage there is that it's potentially more work to build your board (assuming you have the PCB's fabbed by a fab and aren't etching your own or whatever), and it's almost certainly harder to get a reproducible build that behaves the same way.
Altogether, the protoboard and wire approach doesn't scale as well. If you're building some one-off thing for yourself or just building a prototype, it might not matter. But if you're building something that you want tens or hundreds (or more) of, you will want to use a PCB.
The other case where a PCB is clearly better is when you're working with higher frequencies, especially up in the RF range, where you start having to worry about crosstalk / inductive coupling, noise, etc., and minute changes in impedance and capacitance become critical factors. In those cases, it's probably easier to build a PCB that takes those things into account.
Another "non electrical" factor is that a lot of newer components aren't available in old fashioned through-hole DIP packages. If you want to use the latest IC's you may almost be forced to use a PCB, or at least to use a small shim / PCB "adapter" that the SMT component solders to, that lets you treat it as through-hole.