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

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