Could someone explain to me please which aplications demand one and the other and why? As far as I read it's all about the 'db', is it true? And why?
To answer simply - an oscilloscope is an essential tool for any electronics lab, whilst an SA is generally not (unless you are an RF engineer, and even then you need a good scope) and for a good quality one much more expensive in comparison (though Rigol have just brought out some pretty powerful SAs at decent scope type prices)
Basically one does amplitude versus time (scope), and the other does amplitude versus frequency (SA)
(simple) SA example: Say you have a signal and you want to check the harmonic components of it, you can look on the SA screen and check for harmonics (e.g. a pure sine wave should just be one single spike on the screen, at it's frequency, a square wave would be a decreasing series of odd harmonics)
Square wave on a Spectrum Analyser:
The same signal on a scope would look like this:
Scopes typically are digital now or DSO and can be bought from $50 to $5K depending on specs, performance, bandwidth. They can be interfaced on USB, IEEE488, PCI and many other ports. These offer storage for repetitive and 1 shot waveforms and math functions.
Spectrum Analyzers measure Spectral Density and Digital SA's use FFT to calculate the spectrum whereas RF SA's use dual or triple conversion swept scanning like a TV tuner but with very precise preamps, filters and Log converters since measurements are more convenient to display a wide dynamic range such as 100 dB. They are used for seismic, audio, mechanical bearing analyzers in large turbines, radio, microwave, optical spectrum and more. They can be useful for doing Bode plots, filter plots , RF emanation test, Radio tests, antenna design, Radar, Cellular design and test verification.
There are literally thousands of different applications for Spectrum Analyzers besides for Radio Engineers in all fields of Industry where Engineers need to analyze the spectrum in a particular device, whether it is mechanical, optical, or electrical. I know one family relative that use one to analyze Gigawatt GE turbines in Japan for bearing harmonics, which is a strong indicator of product quality and aging factors.
Network Analyzers are even more precise than SA's and have built-in tracking generators with dual inputs so that a transfer Function can be measured. They come in wide ranges of frequency and can be used for measure phase margin in SMPS for stability tests or PLL test or Insertion loss, Return loss , SMith Charts etc. and can be as accurate as 0.1dB from .1 to 50 GHz or a sub-range of interest like 0 ~ 1MHz These can cost $100K each. HP and Anritsu are the two top suppliers in America.
But for plain audio, there are free software tools to display Audio signals and Spectrum Analysis using the MIC, Line IN or internal audio.
e.g. Audacity is one program. I still have the old Cool Edit Pro 2. Version. Waveform Courtesy of AC-DC (Hell's Bells)