Question: Could someone explain to me please which applications demand one or the other and why? As far as I have read it's all about the 'dB'; is that true? And why?

At first I can see Digital Storage Oscilloscopes (DSO) with FFT function and Spectrum Analyzers (SA) as being the same thing...they will get a signal from the Time Domain, and convert it to the Frequency Domain and we can check all the harmonics and frequency components of a signal and analyze it in a whole new way.......But since DSOs usually are much cheaper than SA, I keep wondering what functionalities the SA will offer that a DSO can't. Is it about precision, speed of calculation (my DSO FFT is really slow), bandwidth (cheap DSOs usually go only up to 100MHz), or does it just depend on the models and not on being a DSO or a SA? Is there more that I don't know about and you can tell me?

Answer: 
There were a few correct differences mentioned above, I will try to systemize:

1) Bandwidth (oscilloscope's bandwidth usually wider, but the working band cannot be shifted). I.e. for example Oscilloscope modes are: 0-1kHz, 0-10kHz, 0-50kHz, 0-250kHz, 0-500kHz, 0-2MHz, 0-20MHz,0-100MHz signals, having max sample rate at 500 MSamp/sec. When one looks at FFT, he can see only these 0-100 MHz band.
Spectrum Analyzer may have narrower bandwidth, but it can roll across frequency scale: i.e. for example, bandwidth 40 MHz, sampling frequency 200 MSamp/sec, and working frequencies: 0-6.3 GHz. I.e. Spectrum analyzers modes will be: 0-40MHz,10-50MHz, 20-60MHz, 30-70MHz....6260..6300MHz.
So one can see, that SA has a tunable band filter instead of anti-aliasing LPF in the oscilloscope.

2) Dynamic range. ADC of a spectrum analyzer has a much better resolution.

3) The spectrum analyzer has a low-noise amplifier, oscilloscope doesn't have it. Low noise amplifier, is a special, radio-frequency amplifier, which works in a big range of frequencies, add very low noise to signal.

4) Oscilloscope and spectrum analyzer have different ways to set up triggers. The oscilloscope is oriented on a shape of signal in the time domain, SA is oriented on capturing certain shapes in the frequency domain.

5) The oscilloscope cannot demodulate signals, a spectrum analyzer usually can (because it is virtually an SDR-receiver).

Summarizing: 
an oscilloscope is an extra-wide band millivoltmeter.
spectrum analyzer is a pretty narrow-band receiver, whos main goal to convert radio-waves into the baseband signal (I and Q components) with as low as possible lose and noise.
Another application for a spectrum analyzer is where you'd want to hunt down a source of interference. Latest-gen handhelds make this a lot easier too.
For example, in addition to the spectrogram and standard spectrum analyzer measurements, these instruments can make interference-specific measurements such as carrier/noise (C/N) and carrier/interference (C/I). A trace mathematics (diff mode) can help you find, monitor, and characterize interfering signals.
Another feature is the ability to record spectrum over a specified time. This allows you to find intermittant faults and frequency variations, over time. Great feature. Personally, I'd go for both: Scope + SA. It just makes your bench more useful, long term.