When used literally, both these phrases refer to the objects brought or taken onto a boat, ship or aircraft.
Idiomatically, they have different meanings.
If you 'bring someone on board' , for example in a business, it means that you arrange for them to join your company, team or project.
'To take something on board' is also used in business situations. It means to accept and understand a situation. It's often used to refer to the opinions of others which you need to take into account in your decision-making.