An allowance is a contra asset that is used to reflect expected losses on receivables. These are not identified by any specific receivable but estimated based on past experience. A write-off is a specific reduction against an asset. They both have the same imapct in that the offset is against earnings. However, if a company has an allowance set up in one year that reduced earnings, a write-off would generally serve to simply reduce the allowance previously established and not be a second hit.
For example, if a company has a pool of mortgage receivables, an allowance may be set up to reflect expected losses that not all will be collectible as a % of mortgages over 60 days past due (% scales higher the more overdue a loan is). The gross loan asset is unchanged but a contra asset set up to reduce the value to realizable value. The allowance is taken as a charge to earnings. If a loan then defaults and a portion is uncollectible, then the loan is removed from the asset and the allowance adjusted to reflect the lower expected loss in the future.