但愿 expresses a hopeful wish, often with a slight sense of uncertainty. It's more literary and wistful than 希望 (xīwàng, hope). Often used at the start of a sentence.
culture
The famous line from Su Shi's poem: 但愿人长久，千里共婵娟 (May we all be blessed with longevity, sharing the moonlight across a thousand miles).
