For example, official taxi (i.e. the ones booked via the taxi counter mentioned above) rates from Domodedovo to Central district (which is where Hyatt Ararat is - btw, thanks, Ilja, don't know what I was thinking) range between 3200 and 3800 rubles (or 55 to 65 dollars at current xchange rates).
For the same route, Uber or Gett taxi will be in the range of 1000 to 3000 rubles.
As a reminder - if you use Google maps on your phone, type in your "to" and "from" addresses or locations and the app will provide an estimate for web-enabled taxi services in the area (for Moscow, it is Uber and Gett).
Overall, using the official taxi is a lot more convenient in an airport/train station kind of situation, because your (Uber/Gett/Yandex) driver will need to somehow find you, which will typically involve a call or a text.
In any other setting, web-based taxi services are the way to go.