There is no difference, but the first is more usual. We tend to use the verb 'have' with food, drink, and names of meals. For example, we say 'Have you had breakfast?' rather than 'Have you eaten breakfast?'.
Why? Because it's obvious that you drink coffee. What else can you do with it? As Nathan says, if you say 'drink', it emphasises the act of drinking, which in most cases is unnecessary.
It's not wrong to say 'We drank coffee and ate cake'. It's perfectly clear and grammatical. But 95% of the time a native speaker would say 'We had coffee and cake', because it sounds more natural.