Web Development is very different I feel. Web devs iterate on their product a lot more. A game is shipped. There is nothing after its shipped besides minor patches (if we ignore multiplayer online games)
Web developers constantly need to fulfill new requirements so having a good level of code quality is important.
You might think so, but it really depends on the culture and philosophy of the company.
"Let's rewrite this old messy app in 'modern' technology" - sure it seems cleaner at first, but when you actually reach parity with the old app, you are likely about as messy.
I tend to think that a lot of the messiness in business apps is more due to the complexity of the business rules than the structure of the application.
Web developers constantly need to fulfill new requirements so having a good level of code quality is important.