Are there actually any differences between the two, or is the former the UK term and the latter the US term?
The fungus Agaricus bisporus goes by many names. There seem to be (at least) two cultivars, one of which is white when immature, and the other brown.
When immature and white, this mushroom may be known as common mushroom, white mushroom, button mushroom, cultivated mushroom, table mushroom, and champignon mushroom (or simply champignon).
When immature and brown, it may be known variously as Swiss brown mushroom, Roman brown mushroom, Italian brown mushroom, cremini/crimini mushroom, chestnut mushroom
(Agaricus bisporus, Wikipedia - emphasis added)
That same article also notes that there is another mushroom that's commonly called "chestnut mushroom," Pholiota adiposa.
When I search for "chestnut mushroom," more of the references are to this latter species, Pholiota adiposa, than Agaricus bisporus. So, according to these people, "chestnut mushroom" and "cremini" are not the same.
Personally, I wouldn't put much stock in that BBC article which has two separate entries for "white mushroom" and "closed cap" which appear to describe the same mushroom (Agaricus bisporus). And of course, portabello is also Agaricus bisporus, although, being matured, they might reasonably be considered a different "type."
they are the same mushrooms species.
The white button mushroom is the younger one, the brown/chestnut/cremini is more mature, white portobello mushrooms are the most mature.
I don't see much differences between the white button mushrooms and the chestnut mushrooms as I will cook them down a lot.
For raw applications, I prefer the white ones.
I rarely use portobello, so I cannot comment on them.