Continuous power rating = the amount of power the speaker can handle for a long period of time.
Program power rating = the amount of power the speaker can handle for a short period of time.
With the BW, it can handle a 350 Watt amp, running a sine wave, flat out but with no distortion, and it'll basically do it forever. The program rating means that you can actually use a 700 Watt amp running real music (which has loud bits and quiet bits), because it will only actually hit 700 Watts occasionally. (With a dynamic instrument like a bass, the average power will probably only be about 70 Watts if your loudest peaks are 700)
There are other things to bear in mind though. If you put 500 Watts through the BW at 500Hz, it'll probably be fine. However, 200 Watts at 50Hz will probably kill it through overexcursion. There's only so much bass you can get out of any given speaker, and if it starts to distort you're using too much power, or too much low EQ. Speaker power ratings are a guide to how much power to use (plugging a 1000 Watt amp into a 10 watt speaker is pretty dumb - you'll never get the amp above 1), but at the end of the day if it sounds OK it probably will be OK. If it distorts, turn it down, or get another speaker if you need to be that loud.
Andy