O agregado da consulta SQL pode não aparecer na cláusula WHERE

Use o HAVING conforme indicado pela mensagem de erro, que requer um GROUP BY :

SELECT
    SUM(M1.InvoiceTotal)-SUM(M1.AmountApplied) as PastDueAmount, 
    M1.BillingID, M2.Name, 
    M2.DelinquentDaysThreshold, M2.DelinquentAmountThreshold,
    DATEDIFF(d, MIN(BillingDate),GETDATE()) as DaysLate
FROM
    Invoices M1
    LEFT JOIN
    ClientAccounts M2 ON M1.BillingID = M2.ID
WHERE
    InvoiceTotal <> AmountApplied
    AND
    M2.DelinquentDaysThreshold > DATEDIFF(d, MIN(BillingDate),GETDATE())
GROUP BY
    M1.BillingID, M2.Name, 
    M2.DelinquentDaysThreshold, M2.DelinquentAmountThreshold,
    DATEDIFF(d, MIN(BillingDate),GETDATE())
HAVING
    (SUM(M1.InvoiceTotal)-SUM(M1.AmountApplied)) > M2.DelinquentAmountThreshold