BRP card shows your immigration status.
Once you become British citizen, you become free from immigration control, so BRP has no value.
But since BRP is a physical card, and until it is within its validity date shown on it, it can still be used for whatever purpose you want to use.
If you want to travel out of UK before you apply for British passport, then on your way back you can show your BRP to airline so that they let you board aircraft.
Also not, you do not need to apply for British Passport ever after becoming a British Citizens. British Passport is a travel document, and not having a passport does not mean someone is not a British citizen. There are millions of British people who never had passport and never will have a passport.
Same goes for every country in the world.