Englishman and British man

Solution 1:

It has to do with etymology and usage.

Old English made use of compounds:

Engliscman

The word was in use before, is still today and its spelling has been reformed to Englishman.

There are other words that were in use before and not today:

Englander

The use of the word British may have come into popular use only after, in the 18th Century, when it became interchangeable with English. At the time the English language did not tend to make use of full word compounds that would lead to words like Britishman. Moreover, the word British means related to the Britons (as yellowish means related to yellow) and it would be used as an adjective to British Isles, the people themselves were called Britons.

Similar treats are found in French, while no full-word compounds exist today, words like partout (par tout, lit. "through all", i.e. everywhere), aujourd'hui (au jour d'hui, lit. "at the day of today", i.e. today) and beaucoup (beau coup, lit. "a beautiful hit", i.e. a lot) are still in use.