In order to use TSheets and Time & Budgets in-conjunction, you will need to install custom fields within TSheets to sync with the Roles and Task Types within Karbon.
To install in TSheets, go to Set Up > Add-ons OR Feature Add-ons > Manage Add-ons. Under Get More, find Custom Fields and click Install.
Once you've installed you can begin adding time in Karbon that will carry over to TSheets.
Any additional custom fields added in Tsheets will appear in Karbon's Time Entry modal as additional drop-down menus.