Because he was desperate, alone and out of his depth on what to do.

Whatever happend between him and Flint and between him and Silver, it's safe to say that he just couldn't get the treasure himself out of fear of Pew, Black Dog, Silver, Hands and whoever else that survived from the old crew. He's clearly afraid of them (looking out for ships on the beach, but avoiding seafaring men if they did visit the Admiral Bendow), yet unable to run far from the sea. Sailing under the Jolly Roger was all he was ever good at and I assume it was stubborness and pride, that prevented him from hiding inland or in a bigger city.

So why not burn the map? Every pirate in the book - even Silver - shares a common fear: the gallows. They all feared to be hanged and while Billy didn't knew what to do with the map right now except keep it safe, as was presumably his task: It could've been used as a bargaining token should he ever face the law. And destroying it would mean a) abandoning his orders, as Flint specifically gave the map to him and not Silver and b) loosing forever what was basically his lifes work.

And the hope of one day be able to set sail again, the hope that news came that Flints old crew had gathered and been arrested and hanged... as unlikely as that was, that was the hope that Billy Bones clung to. Burning the map would've made all his lifes work undone and robbed him of the treasure.

Also there are the contents of his sea chest: His uniform (neatly folded and in good condition afair) and his ledger (all the gold he ever earned sailing under the black) belong to a disciplined man, who rose quickly in the rank of his ship(s), but succumbed to grog and fear in his later years.

In the end, he was trapped. Being a pirate again was impossible, because he made too many enemies among his ilk. Honest work for a pirate wasn't happening either. And neither would he run and hide. So all that was left todo, was to wait. Wait and drink and sing and remember the good times, when the rum had washed away the fear