There are three major classes of sound synthesis these days: FM, wave table (software versions, too), and wave guide. You've already learned about FM, so let's take a look at the wave table and wave guide models for a minute.
Wave Table Synthesis
Wave table synthesis is a mix between synthesis and digital recording. It works like this: The wave table has a number of real, sampled digital sounds within it. This data is then processed by a DSP (Digital Signal Processor), which takes the real sample and plays it back at any frequency and amplitude that you need. Hence, you can sample a real piano and then play any note on that piano using wave table synthesis. It sounds almost as good as digital, but you still have to have the original sources sampled. Again, that takes memory. The Creative Labs AWE32 is a good example of this.
In addition to the hardware wave table, there are software synthesizer-based wave table systems, such as the MOD format for Amigas and the DLS system used in DirectMusic. Computers are so fast now that if you just have a D-to-A converter that plays digital sound, you can use it to synthesize digital sound based on software samples of real instruments much like the wave table does. As long as you can make the DSP happen in real-time and can perform frequency, amplitude, and other processing functions, you don't need any hardware! This is exactly how DirectMusic works.
Wave Guide Synthesis
Wave guide synthesis is the ultimate synthesis technology. Through the use of DSP chips and very special hardware, the sound synthesizer can actually generate a mathematical model of an instrument virtually and then simply play it! This may seem like science fiction, but it's a fact. With this technology, the human ear can't perceive the difference between a sampled instrument, the real one, and the wave guide simulated instrument. Thus, you can create MIDI files that control a wave table or wave guide synthesizer and get great results. The Creative Labs AWE64 Gold and greater have this technology.
So the verdict is, a synthesizer can create music as real as real can get, but the musical piece still must be encoded as MIDI. Also, if you want speech or special sound effects, they're hard to do with synthesizers, and even with wave guide technology you'll need special software.
However, with DirectMusic you can program instruments with digitized sounds and play them like notes, so that problem is solved. Thus, you can use digital sound for all your sound effects and DirectMusic for the music. Granted, there may be a little more work involved than just playing a wave file, but DirectMusic sounds the same on all machines, is free, can read standard MIDI files, and has a ton of features if you want to use them. Therefore, you may decide to use a mix of both: DirectSound for sound effects and DirectMusic for music.