It occurred to me the other day that while I'm perfectly comfortable with questions expressed as

"Why can't we do this?"

it sounds very odd to use questions expressed as

"Why cannot we do this?"

Surely "can't" and "cannot" should be equivalent, but they aren't here! Is there some reason why they're different here, or is this just some inexplicable quirk of English?

Probably sounds odd because of the frequency of personal use and use by others of the contraction 'can't'. Things rarely heard do sound odd.

Don't want to awaken the wrath of grammar purists etc who believe that there is such a thing as 'good English' and absolute standards etc. Language and acceptable 'grammar' emerges on the basis of many evolutionary trends - one of these is the contracted form - it's easier to say 'can't, won't, don't and so on)

When people are speaking a language they don't run about thinking 'I must conjugate this verb / decline this noun / enusre that I comply with the rules of grammarians' - they say what is easiest

It’s the way usage has developed. “Why cannot we do this?” would have been quite acceptable a couple of hundred years ago. Jane Austen regularly writes things like:

You and I must establish a musical club, and have regular weekly meetings at your house, or ours. Will not it be a good plan? (from “Emma”).

But somehow our tastes have changed, and we now prefer to say “Will it not be?” or “Why can we not?”.
In the contractions, however, we can’t separate the verb from the “not,” and so we have no alternative to “Won’t it be?” or “Why can’t we?”.

I’m not sure whether “Why cannot we?” is really ungrammatical. I think it’s probably just no longer idiomatic.

It is odd because if you think of it, we don't normally say, 'let us all go to the movies' because it would come off as archaic. If we said in church, 'let's pray' it wouldn't sound quite formal enough!

What if you said, 'Let us dance!' to someone? In my case, they'd be laughing for two reasons because I can't dance worth beans.

In Northern American usage, you don't use the 'shall' much at all or you risk sounding a bit too formal. You probably would use it for 'Shall we go?' however.

I've never heard a North American use shan't unless they were joking. But I've known plenty of English speakers from areas of England who do. Then again, I went to school in the UK for a year, so it's normal that I heard it often.

When I taught English in France, I often had trouble explaining how to translate a suggestion or a command. They would always want to say, 'we go?' to get moving. It sounded very weird but it's logical for them.

I think some of these expressions have evolved over time and haven't necessarily taken the most logical path.

Some languages do use what we consider to be the cumbersome usage. I.e. in German you say "Lass uns gehen" (let us go) for "Let's go." I agree with the consensus that "Why cannot..." is probably quite allowable formally but is just not used because that's the way we talk.

In most spoken varieties of English, it's a feature of some the contracted negative versions that the vowel is different, as in can't versus cannot, won't versus will not, don't versus do not - and of course the oh-so-taboo ain't versus am/are/is not. It seems that some of the contracted negatives use vowels that have in effect become specific markers (indicators) of the negative in these verbs.

I tend to agree with Portobello, the main difference being 'that's how we talk'. In many other ways we write differently and more formally than how we would speak the same expressions.
Expressions such as 'Can't you do that', or 'Haven't you finished', I find quite amusing as what is really meant is 'Can you do that', and 'Have you finished'. Sometimes I'll deliberately answer such 'double negative' questions correctly, with yes or no, meaning 'yes, I haven't finished' or 'no, I can do that'.

TabbyTom's point about what now seem odd or archaic forms of speech having been current and entirely acceptable some time back is very relevant. I could not find the exact sentence but only last week read 'had eat' in Dickens' 'Barnaby Rudge' and as illustrated by Tabbytom, Jane Austen uses many forms no longer 'acceptable'.

My now very elderly aunt who lived in Hawaii once met with President Eisenhower in her house on a golf course out there. She caused great amusement to the great man himself and his Secret Service entourage when, with customary Irish hospitality, she invited him and them to partake of refreshments by saying

'Sure, sit youse all down and I'll give youse a wee cup of coffee in your hand for I'm only after making it.'

Grammarians and lexicographers might well have a field day with parsing and analysis here. Interestingly, there are mixtures here of correct Shakespearean and direct translations of Irish into English. We, in the north of Ireland, retain more than elsewhere some Elizabethan constructions and meanings - e.g. university can be said as 'univarsity' in some areas. This is echoed elsewhere only in deliberately retained forms such as 'Varsity Match'(normally used now in relation to Oxford and Cambridge boat race, rugby matches etc). Some of us would use the word 'fond' not as meaning 'affectionate' but as 'foolish, overly keen, naive. 'You must be fond' a mother might say to her daughter insistent on going out even in the most inclement weather for example. King Lear uses the word exactly like this when he says 'I am a very foolish, fond old man'. Some of us also 'doubt' when we mean 'believe'and 'allow' when we mean suppose. Seamus Heaney says that
'some cherished archaisms are correct Shakespearean' with us.

Sometimes, therefore, and in some areas in particular it would be standard forms that would sound laughable, be regarded as pretentious and / or 'too fancy by half.'

I don't really know why I started on this train of thought but language is fascinating