If we were up very late studying you might say 'we can drink some coffee, that will keep us awake'. The focus here is more on the actual drinking of coffee. If we had not seen each other for a long time and wanted to go somewhere and talk I might say 'meet me here later and we can have some coffee and talk'. In this case we are really just going to spend time together, what we are drinking doesn't matter. I could also say 'we can have lunch', or 'we can have some beers'. These are basically the same. The point is we are going to spend time together, what we are doing is not as important.

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.