Many people use the two terms interchangeably. But if we’re going to get picky about it, we’d say an “effect” treats your audio, in effect adding its own signal to the existing signal (such as a delay effect that adds echos, a reverb effect that adds reverb, etc). A processor changes the entire signal — think a compressor, an EQ, and so on.
However, things have gotten murkier because many engineers like to use compressors and EQs in parallel to the original signal, and only mix some of the processed signal in, just like an effect. And many sound designers will use effects to process the entire signal. So these days, the terms are often combined or used as synonyms.