The service you are referring to is called "dynamic currency conversion" and is sold as an extra convenience in locales with a heavy tourist presence, not just in airports. Many people are tempted to opt for their home currency because they assume that it will relieve them of inter-bank fees and disadvantageous exchange rates.
Starting with your second question...
And what are the benefits of choosing one over the other?
The benefit of paying in local currency is obvious: i.e., you know what the transaction involves. Generally it is picked up by the card issuer and passed through with predictable rates and fees that are governed by the regulator in your home country. So while the end-user faces uncertainty in knowing the precise exchange rate that will appear on their bill, they have the certainty of knowing that it will be set within a regulatory framework and hence 'near the market'.
When you opt to pay in your own currency abroad the exchange rate is established right then by whichever bank is servicing the retailer, and you have no control over it. This can lead to an uncomfortable difference in what you have actually paid.
The regulators in Europe require those offering 'dynamic currency conversion' to exhibit the exchange rate prior to the customer's approving the purchase.
Dynamic currency conversion is legal in the UK and across Europe, as
long as traders display not just the price but also the exchange rate
being used before the payment is made.
Source: Tourists warned over exchange rate costs (BBC, 24 July 2017)
BUT, they do not regulate how the exchange rate is presented to the customer, and therein lies a problem. What may be presented is the inverse rate of the bank's counterparty's bid and this is a rate applicable to the currency wholesale markets and not available to an ordinary mortal. So in order to evaluate it, the customer has to take the inverse of the rate and then compare it to the relative savings they could achieve by using an ATM. People are not generally willing to go through that kind of hassle.
Your other question...
Why are there two options?
Merchants love it because they receive a commission from the DCC provider.
Merchants that operate a global presence, like Costco for example, will avoid offering the service because it interferes with their internal hedging strategies. Amazon, on the other hand, will offer DCC to customers whose credit cards are denominated in multiple currencies; my observation is that Amazon likes to display a bid-side retail rate derived from yesterday's rate (fair enough for most purposes).
This shows a live screen grab from Amazon's DCC. The exchange rate works out to 500 pips over the prior day's mid-market rate. You can't purchase in this quantity (it's too low) from a retail bank for a rate that attractive, so in this case DCC makes sense. Caveat emptor and your mileage may vary.
Extended discussion on the advantages and disadvantages in the DCC Wiki Article.