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