Looks a bit vague question to me -
If I do rename the schema and publish the components again; will the application break ? - You really need to look into your code - in case you have written custom broker queries based on Schema Name or if there is any code using Schema Name for the functionality like Event System code or may be some custom page - This may result in application and/or CMS break.
So you need to look whether there is any dependency in your implementation on the Schema Name.
On the other hand if there is no dependency, you may change the Name of the schema safely from Tridion perspective - refer the answer from Priyank - It is not recommended to change the Xml Root Element name of the schema - because if you change that you may require to synchronize all components followed by a publishing based on that schema plus be sure that it won't work if you try to Content Port from one to another existing environment having this schema with older root element name.
In my opinion, changing just the schema name as per the business naming standard but not the Root Element Name looks weird to me - but yes that may look OK as the business people won't look at the Root Element Name probably and won't understand its purpose as well.