I can't say for any country, but at least for several banks in Russia, the difference is how transactions are processed if they are nominated in different currency than the card [and both are different from the local currency of your bank's country]. E.g. when you pay for something in GBP (and the shop charges in GBP), while your card is in CHF.
For Mastercard, if currencies differ, MC first converts transaction amount (eg GBP) into EUR (using its own exchange rate), then MC charges your bank in EUR -- and your bank converts amount in EUR into currency of the card (eg CHF) using your bank's exchange rate.
For Visa, the scenario is the same except transaction amount is converted by Visa into USD, and USD amount is charged from your bank (which converts USD to currency of the card).
This typically translates into the following recommendation from a bank: "if your card is not in USD and EUR, and you are going to spend more money in Europe, choose MC; if in US, choose Visa".
In my experience, MC and Visa exchange rates typically have much smaller spread ("we buy" / "we sell" ratio) than a typical retail bank--but your mileage may vary. However, the payment system exchange rates are rarely applied on their own, most of the time (if not always) they are followed by bank's exchange.
Bank's conversion rate for card transactions may be different from the rate it uses for other conversion operations (eg for conversion between your accounts, or for exchanging cash at the counter).
For withdrawal at ATM (when currency asked for is different from the card's currency), scenario is not necessarily different between MC and Visa. Eg. for one of the banks in Russia, the amount asked at ATM is first converted into USD by Visa or MC, and then they charge your bank in USD, which converts the amount using bank's own exchange rate.
Update: Here is an answer from another bank again in Russia. Not sure if it contradicts what is said above, or we can just consider it inclusive for above as well (and can remove the answer above).
If transaction currency is different from card currency, AND the currency pair is not what the bank normally performs conversion using its own exchange rates, then:
And if the currency pair is what your bank normally performs conversion for, your bank's exchange rate is applied; no Visa/MC conversion takes place at all.
It still translates into: "if you are going to spend more money in Europe, choose MC; if in US, choose Visa"--but bring more nuances for the rest of situations.