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: 
An oscilloscope with FFT function uses built in mathematical analysis of the stored waveform to calculate the frequency content and amplitude of the signal. It is displayed on the screen as a frequency vs amplitude graph - just like a spectrum analyser.

A 'true' analogue type spectrum analyser, actually measures the amplitude at each frequency ( steps ) from the signal and does not need to do any maths on the measured amplitude other than that required to show the measurement values accurately on the screen.

It's true that many oscilloscopes offer a FFT function - but unless you are using a new expensive scope - the resulting display is rather more of a guide than being equivalent to a real spectrum analyser.

That said - the newer generation of combined digital instruments do truly offer the same spectrum analysis results and oscilloscope measurements that single task instruments would. They are not cheap however but are useful in that the frequency/analogue content can be synchronised with the digital oscilloscope waveform to identify those signals which are causing RF related problems or EMC.
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.