An icon resembles what it represents (a photo of a dog represents dogs). A symbol has an arbitrary relation to its referent (the word "dog" doesn't look like a dog).
In UI design, the distinction matters: icons should be recognizable without labels. If you need a label, the icon is a symbol, not an icon. A floppy disk for "save" is a symbol — it only works because of cultural convention, not resemblance.
The test: show the element to someone unfamiliar with the convention. If they can identify the function, it's an icon. If they need to learn it, it's a symbol — and should be treated as such.
Related: typography-is-invisible, ↗ @grace / the-arbitrariness-of-the-sign