Basically, everything is now an “OV-chipkaart”. There are three main types of cards, all with an RFID chip in it: personal (for people with a discount card, yearly rail card and the like), “anonymous” (for infrequent use) and disposable (for punctual use). Depending on the card, you can load credit and travel using a pay-as-you-go system or directly load a product like a single trip or a day pass that will be valid on a specific trip or specific network. This system is used by the national train company (NS), municipal transit companies like GVB, HTM (the Hague) and RET (Rotterdam), by private train operators like Arriva or Veolia and by other bus companies like Connexxion or Veolia (those two are now part of the same group but not fully merged as far as I know).
With NS, paper tickets have been replaced by single-use chip cards (with a EUR 1 surcharge over regular fares). Those do have an RFID chip too but cannot be used for pay-as-you-go on other companies' networks. The GVB day card is also such a chip card. It's only valid on the GVB network (metro, bus, tramway, and ferries in Amsterdam) but, just like the single trip NS card, it is technically a disposable OV-chipkaart.
Whatever you do, you are supposed to check in and out using an OV-chipkaart card every time you use public transport (incidentally, with day cards, you could imagine the card would be valid nonetheless but I have been told several time that you really need to check in and out…). Even one-hour bus tickets everywhere in the country are actually chip cards (which the driver will often “check in” for you if you look like a tourist). So you have to use some sort of RFID card in any case and, as a visitor, your choice is between using a disposable card for each trip/network or an anonymous chip card for pay-as-you-go on several networks.
If you choose to use disposable NS tickets and a GVB day card, you don't need to get anything else. But if you do get an anonymous OV-chipkaart and load it with enough credit to travel on the train, you might just as well use it on the Amsterdam local transit network and you wouldn't need the GVB card. It's up to you.
In terms of costs, it's going to depend on exactly where you want to go. With NS, you have to pay some surcharge anyway and the biggest problem is that you need to have at least EUR 20.00 on the card when checking in for pay-as-you-go. So for a short trip like Amsterdam CS-Schiphol (where the regular fare is EUR 4.00), you would have at least EUR 16.00 tied into the card when arriving at the airport, which is annoying.
If you don't travel a lot in the city, paying single trips with an anonymous chip card would probably be cheaper than a multi-day card (you pay per kilometer) but not by much. The main drawback is that the anonymous non-disposable chip card must be purchased for EUR 7.50 (before loading any credit on it) and you would then need to deal with the whole pay-as-you-go system.
One variant would be to buy an anonymous card, load it with EUR 20 and use that for the way from Schiphol to the center and then in the city. After four days, you would have used most of your credit and probably spent less on the GVB network than the price of a GVB multi-day card (at least if you don't travel all day). Then you could buy a disposable NS ticket for EUR 5.50 (that's EUR 4.00 with disposable card surcharge and credit card fee) for the transfer to the airport to avoid loading too much credit on your anonymous chip card at the end of the trip.
But at the end of the day, no matter what you do, the total cost will be over EUR 30 and the difference between anonymous chip card and GVB day card is not going to be large. What you must definitely avoid is paying the trip to the airport with pay-as-you-go (except if you intend to reuse the card for a future trip or intend to give it away to someone else who can use the credit).
The easiest solution is therefore probably to buy single-use train tickets from the NS machines (for transfer to and from the airport) and then use a GVB multi-day card for all your journeys within the city.
Assuming two ~3 km trips in the city a day (to fix ideas that's approximately the distance between the Central train station and the Museumplein), the costs would be:
- Single-use NS tickets and GVB day card: EUR 5.50 for each train trip + EUR 21.00 for the GVB card. Total: EUR 32.00
- Anonymous chip card and single-use ticket for the return to Schiphol: EUR 7.50 + EUR 0.50 + EUR 4.00 for the way from the airport to the city center + EUR 11.00 for tram/bus + EUR 5.00 tied into the card at the end + EUR 5.50 for the transfer to the airport. Total: EUR 33.50.
- Anonymous chip card only: EUR 7.50 + EUR 0.50 + EUR 8.00 for the train + EUR 11 for tram/bus journeys + EUR 16 EUR tied into the card at the end. Total: EUR 43.00
- Disposable cards all the way (buying train tickets with credit card and bus tickets with cash each time you step into a bus or tram): EUR 5.50 for each train journey and EUR 2.80 for each tram/bus journey. Total: 66.40 EUR
Also, it's now possible to load credit on an OV-chipkaart with foreign cards in all train stations but NS applies a EUR 0.50 surcharge for credit card transactions.