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.
Current day spectrum analyzers (SA) are seldom fully sweep tune. Most do FFT and stitch channels together to form a frequency span.

Besides a class of modern SA measurement such as Vector Signal Analysis, does not stitch channels, but rather, measure the entire channels base on the IF sampling rate. The analysis bandwidth, which is is usually around [IF sampling rate/1.25] is up to 1 GHz, for the highest end SA -- [Keysight UXA](http://www.keysight.com/en/pdx-x202152-pn-N9040B/uxa-signal-analyzer-multi-touch-3-hz-to-50-ghz?cc=US&lc=eng). 

Non exhaustive of scope vs spectrum

1. Scope digitize from baseband to the desire frequency range. SA downcovert the RF signals and digitize at IF
2. Being able to digitize at IF allow SA to have better vertical resolution. A scope vertical resolution is mostly 8 bit, while SA is up to 14 bit. (Digitizer designers trade sampling rate with vertical resolution)
3. A scope is useful for time domain analysis. A spectrum is better for frequency domain analysis. SA having a better vertical resolution will have a better performance in S/N ratio, allowing one to see signal at very low power level. While scope having higher sampling rate will allow better time resolution of certain kind of measurement such as rise time.
4. A scope can be more than one port while SA is one port. Hence a scope is able to perform multi channel time domain comparison such as phase, pulse rise time...etc

[![](https://i.stack.imgur.com/dBHHP.png)](https://i.stack.imgur.com/dBHHP.png)
Above: Scope measuring multi-channel pulses