In my opinion, this is the same question as whether or not to use Arch Linux over another distro. Roll your own simply if you want, or if no other synthesizer is doing it for you. Sometimes it's just worth doing it for the fun of it, or for the sake of learning. I'm actually planning on making a bunch of soft synths, so if you like, you can let me know what you'd like to see.
I think immediately of libao for the C programming language. Imagine a while loop that completes itself at least 44100 times in a second. If the synth cannot write to the buffer that quickly, the sound card runs out of samples, and can't do anything, so it stops playing. Hence that clicky mess sound. This is for realtime synthesis though. If you can produce audio at your own pace, you have the opportunity to sound good every time.
Fantastic question! A synthesizer is a device that generates audio signals. I remember reading somewhere that they were sometimes referred to as "noise machines", in regards to I think the Minimoog specifically. A drum machine is a type of synthesizer, as were the electronic-keyboard-having synths of the past.
Thank you so much for the offer! I'll mostly stick to soft synths, but I don't mind going into the history a little bit to explain terminology and whatnot. There's a surprising amount of overlap between analog subtractive synths and software subtractive synths anyways.
Excellent point, it's easy to forget that sometimes. What makes a good gui, in your opinion?