When a cost or payment is prorated, it's proportional to the amount of time spent doing something, while pro rata describes anything distributed in a fair way, according to a specific proportion or percentage of a whole.
Prorated is from the Latin pro rata, "in proportion." If the lease on your new place starts on the first of July, but you move in on the 15th, your first month's rent may be prorated so you only pay for the two weeks you lived in the apartment. Similarly, the initial paycheck for a job begun mid-month is prorated, compensating you just for the days you actually worked:
"But his pay was prorated in 2023 because he didn’t join MLS until midseason." (Los Angeles Times)
"The building had backup generators, but there was still cold water and intermittent power for 9 days; the landlord offered tenants a prorated rent reduction to compensate for the inconvenience." (The New York Times)
Pro rata is the root of prorated, but it's also a separate term with a slightly different meaning. It refers to a proportional fraction of a total, like pro rata dividends paid to company shareholders, with the dollar amount fairly reflecting the percentage of the company they own:
The first $35 million from the settlement will be distributed pro rata, for an average of about $165 per loan. (Reuters)
"Investors will also receive a pro rata share of the existing assets of the entity in the event of a liquidation." (The Wall Street Journal)
In everyday language, prorated and pro rata are often used interchangeably to mean "proportional." If you want to be more precise, consider whether you're describing a share of something relative to a whole. If so, think ratio and use pro rata. But if you're talking about an adjustment in cost or pay based on length of time, the better word to use is prorated.