When I have a spectrum analyzer, do I need a power analyzer?
Yes and no. A spectrum analyzer can do some things a power meter can but not everything.
Mainly it can't measure the total power in a given instant since it sweeps the frequency. Also a power meter is way more rugged and usually it has more power handling (there are two types of spectrum analyzer users: these who fried the input mixer and the ones which doesn't… yet).
If you do testing in CW in most cases a spectrum analyzer could do the job.
EDIT: usually a power meter is more accurate since it's an integrating device, but that's a finer point
EDIT: referring to power handling: a spectrum analyzer is designed to handle typically milliwatt level signal (remember, the input is terminated on 50 ohm so everything is dissipated on that terminator).
For high power you need an attenuator (colloquially called a 'bird' from a manufacturer name), or a power splitting network (and the associated complexity).
Since in the comment you're talking about 25W of power we are handling something like 45dBm of power with a spectrum analyzer you most likely overload it (often the max is 0dBm, the value is always printed over the input connector). 25W seems a lot but is actually one of the smaller power attenuators, you'll find them in the kW range too…
As an aside you'll probably need a VSWR bridge (reflection bridge) too, to test antennas in TX mode.