What is the difference between BS Zelda and the original Zelda1?

BS stands for Broadcast Satellite (not "Broadcast Satellaview", which is a common mistake). The Satellaview was an interesting piece of kit - a satellite modem for the Super Famicom, released only in Japan.

BS Zelda no Densetsu (BS ゼルダの伝説) was a rewritten version of the original Legend of Zelda, downloadable from the satellite in four installments. Installments were broadcast at a specific time, because the game featured a live voice narration that at intervals offered gameplay clues over streaming audio! It also had a real-time clock that triggered specific events.

The game has the same basic mechanics as the original, but features entirely different dungeons. There are some minor changes to the layout of the Overworld and the Rupee limit, and Link is replaced by a choice of male or female Satellaview mascots (a boy with green hair and a backward baseball cap, a girl with red hair).

See also Wikipedia, which has a small amount of additional information.