'Cold' is a lower temperature than 'cool'.  In order of rising temperature, we have:

Cold

Cool

Warm

Hot

'Cool' is also more positive than 'cold'.  It's not pleasant to feel cold, but it is good to feel cool - especially when the weather is hot. For example:

On hot days, I like to sit under the trees where it's nice and cool.

Cool is less cold than cold.The weather is cool today- not hot

The weather is cold today -colder than cool.