FM and additive synthesis have almost nothing in common. The only common aspect I can think of is that many FM synths allow using several oscillators or "operators" summed i.e. mixed together in adjustable mixture ratios, which is not in principle even a part of FM synthesis. The thing that FM synthesis conceptually does not do is add waves together, even though real-life FM synthesizers allow for that to some extent.
In FM synthesis, the frequency of an oscillator is modulated by another audio rate signal.
Think about the tuning knob of a guitar. Pluck a string and start turning its tuning knob back and forth - the string's pitch goes up and down. You are modulating a frequency. If you could do this up/down tuning at a rate of, say, 1000 times a second, and if the guitar could actually do this, you could say that you're doing frequency modulation. Or actually, if one string's oscillating wave motion could turn the tuning knob of another string, you would have an FM guitar.
The set of frequencies created by the modulation depends on the accuracy of the oscillators and operations involved, and the precision needs to be very high so that it creates the same frequencies reliably enough for musical applications. That's why in practice, frequency modulation got very popular only when it was done with digital circuits which allow for mathematically perfect accuracy and timing. By perfect I mean that digital calculations always produce exactly the same numerical output for the same input, and with discrete timing, so the operations produce stable results that are not subject to the analog world's tuning instability.
The sound created in FM, when you can modulate an oscillator at an audio rate is not at all intuitive. I don't think anyone could imagine the sounds created by FM synthesis - they're very surprising, rich and complex, and the amount of overtones added can be very large. Small changes of parameters and "algorithms" in FM synthesis can change the sound dramatically like in quantum leaps. This is why some people see FM synthesizers as wild beasts they can't control.
In additive synthesis, sine waves are added together. https://en.wikipedia.org/wiki/Additive_synthesis
It's a bit like a Hammond organ or church organ where you can create a mix of overtones/partials/multiples of a fundamental base frequency. This is a very intuitive and simple way of molding sounds. You can dial in more or less highs and lows etc.