In the US, a subsidy is given to you by the government for specific purposes. It can be a discount, like subsidized insurance, or it can be direct monetary payments, like childcare subsidies or farm subsidies. Money given to the poor for food and housing is called "welfare". Meanwhile, parents give children an "allowance"--a small amount of money that they get every week and which they can use however they want. In the past, when men worked and women stayed home, men gave their wife an "allowance," but now that would sound offensive, even in families where only one person works. I think I have seen the words used differently in books from England, so see what other posters have to say about British usage.

subsidy is same as discount while allowance is cash or money given to you for upkeep.