Sound synthesis generates new sound; signal processing modifies an existing sound
Sound synthesis is the electronic generation of sound from scratch — oscillators, noise generators, or algorithms produce audio signals without a pre-existing source. Signal processing is the modification of an already-existing sound (a microphone input, an audio file, or a synthesized signal): applying filters, reverb, distortion, etc. In practice a synthesis system also does signal processing (filtering its own oscillator output), so the distinction is about the primary source. Understanding the split matters because synthesis requires designing a signal from nothing, while signal processing requires knowing the spectral content of the source before deciding how to modify it.
Examples
Additive synthesis: summing sine wave oscillators to build a complex timbre from scratch (synthesis). Applying an EQ to a guitar recording to cut low-end rumble (signal processing).
Assessment
Classify each action as synthesis or signal processing: (a) running white noise through a bandpass filter; (b) modulating a sine wave’s frequency with an LFO; (c) adding reverb to a field recording.