Allowance and 【Concession】 both signify a change made by way of compromise or adjustment.
Allowance usually implies a modification or variation of a requirement or a standard made for a good reason (as probable contingencies or mitigating circumstances).
make 【allowance 】 for the current in steering toward the opposite bank
make 【allowance 】 for his inexperience
make 【allowance 】 for wear through friction in designing the parts of a machine
if business imposes its restraints and its silences and impediments, Mr. Darnay as a young gentleman of generosity knows* how to make 【allowance 】 for that circumstance —Dickens
【Concession】 implies that the change has been made reluctantly and usually as a favor or indulgence.
they would make no 【concession】 to a candidate's youth and inexperience
the sole 【concession】 to leisure allowed me out of the year was one month on a farm —Heiser
any 【concession】 to fashion was, they felt, unbecoming to their age —Sackville-West