Do you say 'thank you' or 'thanks'?

'Thank you' is formal and 'thanks' is informal.
However, both 'thank you' and 'thanks' are polite ways of thanking people and accepting something that somebody has offered you:
'Will you join us for dinner?' 'Thank you very much!'
'Would you like a coffee?' 'Oh, thanks.'
'How are you?' 'I'm very well, thanks / thank you.'
'How's everything with you?' 'Fine, thanks.'
'Thanks a lot.' 'Not at all.'
'Thank you very much.' 'You're welcome, sir. Have a nice day.'

Also, they are both used to show that you are grateful to somebody for something they have done (Thanks / Thank you for something or for doing something):
'Thanks a lot for your help.'
'Thanks for all your help. 'Don't mention it.'
'Thanks for doing that.' 'It's a pleasure.'
'Thanks for the ride.' 'No problem.'
'Thank you for your generous gift'
'Thank you for all your help.'
'Thank you for being so honest with me.'
'Thank you very much for the flowers.'
'Thank you for your warm welcome.'