They're basically the same, and often used interchangeably. However, the perspective is the direction of the action: If you're on the ship, you can ask "What did you bring on board with you?" and you can ask "What will you take ashore with you?"

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.