The deciding factor between stand-alone analyzer or soundcard would be the frequencies you're interested in.
I will record the sound it makes with a microphone (it sounds differently if it is calibrated correctly than if it is not),
If the difference is audible, this means the frequencies you want to measure are lower than 20kHz, which means audio gear will work. This is good news as it's readily available and cheap.
Don't use a RF spectrum analyzer for this! First, there is no need to pay for extra bandwidth you don't need, and second they have lower bandwidth limits that may not suit you. There is no need for specialized equipment either.
A soundcard (or audio input jack) on a PC, a tablet or a phone should work fine. If you have noise issues, the most likely culprit would be a cheapo microphone preamp inside your hardware, so consider an external preamp, or a microphone with integrated preamp. Try the cheapest solution first!
Your problem will most likely be how to get a good signal, which means select the right microphone/sensor, preamp, and technique.
If the sound is airborne, then a mic will work. However if you're interested in vibrations (say, to check on a bearing) then a piezo transducer or accelerometer applied to the chassis of the machine will pick up vibrations a lot better, while ignoring airborne noise. Why not a guitar piezo pickup? Or perhaps a directional microphone. Or two mics and take the difference between the signals. Good acquisition is essential, a clean signal is much easier to work with.
So think about where the sound you want to acquire comes from, where the noise you want to ignore comes from, and decide on which sensor you need.
There are several nice software packages to display real time FFT on a PC, try visual analyzer (free) for example. If you can see differences in spectrum then you can write a bit of software to acquire the signal, FFT it, and detect the relevant stuff automatically. I'd use python+scipy for this, as it has powerful and convenient signal processing tools. Or exploit a guitar tuning app as Henry says.