You need to link the two tables together. Create a new Link to a record field in [Live Expenses] that links to the [Budget] table. For each record in [Live Expenses], link to the corresponding category in the [Budget].
In the [Budget] table, you will now see that there is a new field with links to the [Live Expenses] table. There may be multiple expenses for some rows.
In the [Budget] table, create a new Rollup field that combines the data for all the related expenses. Have that rollup pick use the field in the [Live Expenses] table that has the expense amount. You will probably want to use the SUM aggregation formula here.
The [Budget] table should now contain two numbers–the budget number, and the rollup of the live expenses. You can compare them in a formula field.
Keep in mind that this method does not take into account that most budgets are monthly. You will need to find a way to deal with having a budget for each month.