cool — cold
“Cool” (for the right pronunciation, please click here)

cool weather
is, mostly with a positive connotation (sense), ‘a pleasant feeling of weather or a drink that is neither warm nor cold; the enjoyable level or degree of cold weather; (of a person or animal) calm self-control, (of person) not very friendly, (of things or people) excellent, attractive, good to use, etc. ’,
e.g.
We went for a long walk in the open fields enjoying the cool evening breeze.
You look cool in your new dress, darling!
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“Cold” (for the right pronunciation, please click here)

cold weather
is usually with a negative connotation (sense)
(a) ‘low temperature of weather that is unpleasant and uncomfortable’,
(b) ‘a kind of illness with a running nose and a headache — common cold’,
(c) (of a person) ‘not very friendly’, (of a person or animal) ‘not having sexual desire’,

suffering from a cold
e.g.
We returned early from our evening walk because it was getting dark and cold.
The old lady caught a cold when she went out in the cold weather.
