Number 1 is correct 99% of the time. Number 2 could be correct if you were anthropomorphizing (talking about an inanimate object as if it were a person) the weather but only really if you were doing some creative writing.

1 is correct. "I am depressed (because of the weather)". A person is or feels "depressed". A thing is "depressing". A depressing thing makes a person feel depressed.

Grammatically, they are both correct, but only the first one has a meaning. The first sentence suggests that the weather is cold, grey and gloomy, and so it is making me feel unhappy. The second sentence doesn't mean anything. If you are talking about actual weather systems, rather than the way that the weather makes you feel, you could use the word 'depression'. For example, a weather forecaster on TV might say 'There is a depression over the south of the country' or more commonly 'There is low pressure ..'.

Dear Jin, Number 1 is correct.We use depressed for human being.

Today's weather is depressing, yesterday's weather WAS depressED. "is" and "-ing" is present time and "was" and "-ed" is in the past and already happened.