I needed three different car rentals in May for three different places.I got Auto Europe from this site and discovered that it was cheaper to use the Europe site and not the USA one,It was cheaper to rent in Euros than USD.The main difference was the exchange rate I saw.I looked at a car for three days that was 98 Euros.For fun I put USA as my residence and booked the same car which came in as US $133.Using a currency app 98 Euros should be around 102 Euros!
This is just a classic example of why it is written so many times here that you should always do transactions in the local currency. Your bank network gives you the network exchange rate. When you use your home currency the merchant will charge a jacked up exchange rate, because he can. Similarly any cash transactions you do need to be in the local currency, you will be charged a premium for using US dollars that will also cost the merchant to deposit and exchange.
Although Larry's right that you can lose considerable money by opting to pay in dollars on some websites (and in person whe using credit cards), it is also true that when companies have multiple websites, their products and services may be priced differently on the various platforms.
In which case it would if all possible be best to use the web site that is native to the location involved.
It can go either way. I've seen car rentals be cheaper on the European website, and I've seen them cheaper on the US site. And it's often difficult to override the browser's defaulting to the US site.