Generally speaking, what is the difference between these two? What I mean by that is: in what kind of different processes are these produced (and used in physics)?
Plane wave has infinite extent and is an idealization that cannot actually occur in the theory. The wave packet is limited in extent and is more realistic description according to the theory.
A 'packet wave' will consist of a range of frequencies, around the nominal frequency. A continuous wave (sinusoid) will, in the limit, consist of just one frequency.
Why are plane waves said to be infinite in extent? Why cant a wave have one frequency without being infinite in its "domain"?
It's a matter of definition. If the wave doesn't extend to infinity, it must be 'modulated' to localise it. If you modulate a continuous sinusoid, you introduce other frequencies (sidebands).
Because in "plane wave" you have the adjective "plane", which is by its meaning infinite in extent. Frequencies have nothing to do with "plane wave". There are plane waves that have non-periodic profile, like ones used to model shock wave - when analyzed into Fourier integral, all possible frequencies are present.
Btw, that Fourier business works for spatial truncation too. That accounts for diffraction at a boundary.