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Did your team win the football match yesterday?

The weather was very bad, so we weren't playing.

vs

The weather was very bad, so we didn't play.

Is the first one not correct? Why is it?

Okay, so this is a very specific answer to a very specific question. In this specific case, where there are sports teams involved, there's actually a difference in how I, as a native speaker of British English, understand those terms.

We weren't playing.

This means that the team wasn't scheduled to play. So, in the case of the weather, you might say:

The weather was very bad, but we weren't playing anyway so it didn't really matter.

On the other hand, we have the meaning you want in the other option:

We didn't play.

That means that you didn't play. Playing was not done by you. For any reason, you didn't play. In certain contexts, it might have been that you were supposed to play but didn't, or it could mean you were never scheduled. Thus we get to your example sentence:

The weather was very bad, so we didn't play.

This means what you want - you were supposed to be playing, but the weather was bad so you didn't.

Now, the fact you have the so in the weren't playing example means that native speakers will understand what you meant. You've indicated a causal link between the two statements, and thus you must mean that you had been intending to play and then didn't. However, it's more idiomatic, in my experience, to use we didn't play in that situation.

I've no idea if this is unique to British English or not.

Complementing the British English perspective provided by SamBC, here's an American English perspective (more specifically Midwestern American English).

There's a slight linguistic difference between the two forms. 'Did not play.' uses the past tense of 'to do', while 'Were not playing' uses the past tense of 'to be'. In many cases in casual English, there is little semantic difference between these two verbs other than making sure the rest of the sentence has the correct structure ('do' always¹ pairs with a present tense verb, 'be' always¹ pairs with an infinitive form).

Idiomatically, in informal American English, the two example responses are mostly interchangeable.

In formal American English, however, there is a potentially important difference.

In formal usage, the first example expresses through the use of 'were' that at a specific time in the past, the act of 'playing' was not being carried out by the group of people referred to by the pronoun 'we'. In contrast, the second example expresses through the use of 'did' that a specific past instance of the act of 'playing' was not carried out by the group of people referred to by the pronoun 'we'.

This difference leads in turn to most of what dictates preference in American English for which form (using 'were' or 'did') 'sounds' right, independent of whether it's formal or informal usage. In your particular case, the second form ('we didn't even play') is what most native speakers here in the US would pick when responding to the question, because the question is asking about a specific event. For the same reason, the first example, sounds a bit odd, at least by Midwestern standards of American English.

For what it's worth, you can blame this peculiarity of English on it's roots in Old French. Other Germanic languages use a single form for both meanings (without a couplar verb such as 'be' or 'do'), instead inferring this differentiation from context.

¹ 'Always', of course, having the usual 'almost all the time except when dealing with a small handful of truly random words that no sane person should have to remember' meaning that it typically does when discussing the English language.