'Can' and 'could' can both be present tense. 'Could' sounds more indirect, like you're saying you could only do something IF something else happens. When making a request it sounds politer than 'can', as it does in your example. Using 'could' in a request suggests you're omitting 'if you want to'. In the past tense, you can only use 'could'.

In that sentence it means the same thing. Can is more of a direct question. You are asking if you can stay with them. "maybe I could stay with you" might confuse the person as to whether you are asking them, or deciding if you want to or not.