It appears it was calculated using different days. The per diem amount was correct but the old company calculated counting the day issued to first of month which resulted in an extra day being counted.
^^^Sounds correct to me. Weren't you collecting interest for the day of closing?
Besides, as Randy pointed out, it's a COMPARISON table not a match it up exactly table.
The new vendor should do what you tell them to do unless they can prove via citation that you are doing it incorrectly because ultimately YOUR BANK is responsible for what's on the docs, not any third party vendor.