What's the difference between 房子 and 屋?
屋:

In standard Mandarin, 屋 is not commonly used as a noun by itself. Though, 房屋 is a common word, which is very similar to real estate in English. e.g. 房屋中介 = Real estate agency.

Another common usage of 屋 is in forming a name of a special kind of house. In such usage, it normally has a quite cozy feeling, e.g. 小屋 is a hut, 淑女屋 is a female clothing brand name.

I get feeling that the use of 屋 was may be introduced from Japan in modern times. Just a wild guess because 屋 seems much more common in Japanese.

房子:

It is quite widely used, and is very similar to the English word "house".
E.g. 我有一幢房子 = I have a house. 老房子 = Old house, etc.

In general, 屋 is not as widely used in Mandarin as 房子. And 屋 is used in a more formal fashion compared with 房子.