○屋 is used for a shop that sells ○, but the meaning of 展 is not shop, but roof, like in 部屋、屋上、屋代、屋根
店 on the other hand does mean shop, and is used in words like 店内, 店員, 店長.
Also, while 店(みせ) or お店 are single words, 屋(や) is not used alone as a word .
As a suffix, ~屋 is way more productive tran ~店. ~屋 can be used to talk about the shop owner instead of only the shop itself. Also, as ~店(てん）is onyomi, it would be felt as more formal.
屋 is used for types of shops and the person doing the job like 肉屋 (meat shop, butcher), whereas 店 refers to the physical shop and its location as in 東京店 (Tokyo subsidiary) or words like 開店・閉店 (opening / closing up shop)
