Re-running the bill can pick up changes or reflect new account configuratinos that were otherwise not captured previously.
As with any other bill item in CloudHealth, when you re-run a historical bill you are manually kicking off the job that re-applies rate cards, re-applies billing rules, and potentially pulls an updated bill source from Microsoft (CloudHealth will pull new rate cards from Azure, and so we are reliant on the information provide by Microsoft)
These are all actions that can change a historical bill.
Technical Support will not collect historical bills unless directly requested via a support ticket.