What's the difference between "briskly" and "crisply"? Why is "briskly" correct in "We started off walking briskly" but "crisply" isn't?
They're different words, with different meanings. There's a strong collocational relationship between 'briskly' and 'walking'. There's a strong collocational relationship between - for example - 'crisply' and 'starched'. The words 'briskly' and 'crisply' don't overlap very much in meaning (except maybe in the context of military behaviour: a crisp salute tends to go with a brisk march).
So 'briskly' is correct in 'We started off walking briskly' because of that collocation; 'walking crisply' is hardly ever right (except perhaps of someone walking through crisp snow).
brisk: Definition, Synonyms and Much More from Answers.com
crisp: Definition and Much More from Answers.com