A symbol stands for something.
For example, a heart can stand for love.
A cross for Christianity.
They can be physical things (a drawing of a heart or an actual cross) or they can be used by authors to mean love or Christianity.
Whereas a metaphor, is just an image.
Here is a famous poem by Robert Burns, the great Scottish poet:
A Red, Red Rose
O my Luve is like a red, red rose
That’s newly sprung in June;
O my Luve is like the melody
That’s sweetly played in tune.
He is calling his love, the person he loves, a red rose. That is a simile: he compares his love to the red rose.
Whereas, if you say: love is a rose, that is using the image of a rose to describe love. It is a metaphor. Or a symbol, depending on your interpretation.
However, you could write a book and the roses could be symbols of love or just about anything.
Umberto Eco's very famous book The Name of the Rose is a symbol, according to the author himself, but he really does not say of what. Wikipedia goes into the subject rather well:
The Name of the Rose
The point is that the same word can be used as a metaphor or as a symbol. Everything depends on context. Is an author using a thing to represent something else or is an author using a word (or phrase) merely as an image? That is the crucial question one must ask in deciding.