Do I need a different bottom bracket?
The datasheet for the crankset says it calls for a 123 mm spindle length. This would mean that your crankset would sit about a half centimeter further out than the existing one if you use the Shimano recommended bottom bracket, which will affect the chainline. 

I'd go for trying it out as is -- it will probably work. If the shifting is fine, and the chain isn't jumping or derailing when its not supposed to, you're good to go. If it is having problems after adjustments, get a new bottom bracket.