There are several ways of doing FM as well as a few different approaches to additive synthesis.
In the simplest form of two oscillator FM, a sinusoid modulates the frequency of the second sinusoidal oscillator. Depending on the ratio of the modulator to carrier frequency and the modulation index (strength of modulation), you get different sounds. Simple integer ratios such as 1:2 or 3:2 produce harmonic spectra, irrational ratios yield inharmonic spectra. The partials' amplitudes depend on the modulation index, and cannot be controlled independently. There are extensions to more than one modulator or carrier, cascaded FM and feedback FM, all of which extend the possibilities in various ways. For a very easy introduction to FM, try to find a copy of FM Theory & Applications by Chowning and Bristow.
Additive synthesis builds up sounds from sums of sinusoids with given frequencies and amplitudes that may vary over time. The advantage of additive synthesis is that any sound whatsoever can be represented. By doing a Fourier analysis of short windows of the sound you get a series of time frames of the spectrum at that moment. From that representation you would usually pick the strongest spectral peaks and synthesize the sound from a sum of sinusoids. This technique is known as the tracking phase vocoder. A popular extension is Spectral Modelling Synthesis (SMS) which incorporates noise.
Whereas additive synthesis has its corresponding analysis technique that lets you resynthesize (and modify) any sound, the situation is more difficult with FM. Traditionally, FM algorithms were tweaked by ear and by intuition. Then some researchers began searching for synthesis parameters that would match given sounds by genetic algorithms. That approach has mostly focused on harmonic sounds. In the early years of computer music efficiency concerns favoured FM over additive synthesis, but developments in algorithms and hardware makes that point moot.
Both FM and additive synthesis are useful for sound design, but there are of course other approaches worth knowing about, not least subtractive synthesis and physical modelling.