It all depends on the situation, the cause of the problem and the extent of it. In my opinion, if the client has suffered due to the vendor mistake, best would be accept it and try to find out the next best option to overcome and dilute the damage done. 

There are many things that can be done to bring down the loss

1. If you are an airline company, then free fly miles or a complementary meal or a complementary ticket can be given

2. If you are in software then some discount can be given

3. Apology emails or letters can be given with some coupon like food coupons or travel coupons 

Accepting the mistake is a big step and the most important one. Accept your mistake and extend your apology. Then you work on fixing the damage and if possible going an extra mile to patch up for it through compliments or discounts or if none then an assurance of not repeating the same in future. Every mistake can be taken as a learning experience. It is crucial to have a good connect with your client and so always be on your toes to maintain it.