On board describes that something is aboard a vessel, i.e., the location of something or someone.

Onboard is one word (sometimes hyphenated—on-board) when it comes before the noun it modifies (e.g., onboard radio, onboard computer). Elsewhere, writers usually make on board two words. For instance, one might write, “We brought a radio on board so we could have an onboard radio.” Reference: "Onboard vs. on board", from grammarist.com

It's rather an idiomatic phrase (or a quasi-adverb).

Examples:

• There are no medical physicians on board.
• Smoking is not allowed on board.

Aboard modifies an action and is often followed by reference of the vessel, e.g. to step aboard a something.

Grammarist supplies several examples (same reference as above):

Examples

Onboard
Justin boasts an onboard 3-D camera system for analyzing points in space. [Wired News]
Although the airline grabs headlines for threatening to charge people to use onboard toilets or save money by dumping co-pilots, it normally turns to conventional ruses. [Guardian]
At one highway fill-up, the onboard computer showed I had a range of 880 km. [National Post]

On board
President John F. Kennedy called Shepard after he was taken on board the aircraft carrier that retrieved him from the ocean. [USA Today]
San Pietro was being sailed by the remaining crew on board. [Stuff.co.nz]

"aboard" is only a contraction of "on board". May be pilots prefer "on board", but actually there is no difference between the two variants.

We are on board the plane because we have boarded the aircraft.

Aboard the ship, we watched the waves.

Perhaps a bit silly, but:

As you are now on board, welcome aboard the our vessel.

As you are now aboard (the vessel) let me say welcome on board!

I don't see any grammitical difference, rather it seems to be usage as Fumble (hi ya Fumble :) says. (And there's Amid ship - center of ship.)

It would be even more silly, but grammatically correct, to say:

As you are now aboard, let me say welcome aboard.

I wonder if the word 'board' derives from the planks / boards used from dock to ship to enter the vessel?

"Arrr matey, Howard, walk the plank off board!"