From my limited knowledge, it looks like they just sat around for a thousand years before the Ottomans wiped them out. Many Slavic nations were adopted their religion and a modified version of their alphabet, but what aside from that? What were their contributions to art and science?
it looks like they just sat around for a thousand years
If by "sat around for a thousand years" you mean "fought a thousand-year struggle for their very survival".
The Byzantine Empire had, for most of the medieval period, the most advanced culture in Europe, and was extremely influential beyond its borders. You will find that most of the kingdoms and empires of Europe and the Middle East emulated the Byzantines in many respects, for example: the double-headed eagle of the Holy Roman Empire, the Russian Empire, and various Balkan states of today are all derived from the original Byzantine double-headed eagle, the Turkish star and crescent flag is derived from the symbol for Constantinople. These things are but a taste of the much wider cultural legacy of the Empire.
For a list of Byzantine achievements, I invite you to take a look at this list that I have compiled.
For a specific discussion dedicated to Byzantine scientific, educational, and philosophical contributions, see here.
Let me know if you have any further questions.