The express trains from Heathrow and Gatwick are certainly the fastest routes, but not the cheapest, if your hotel is close to the stations in London. Otherwise, you will then need to transfer to the regular Underground or take a taxi, depending upon your hotel's location and how much luggage you have. That will add to both the cost and the time.
The Piccadilly Line tube service from Heathrow or the regular (non-express) train from Gatwick are much cheaper and can often be more convenient, especially if your hotel is near a station. If you have a lot of luggage, you also need to be aware if there are a lot of stairs coming out of a particular station, especially if taking the Underground. Transport for London publishes a useful guide for the accessibility at each station. A TfL Oyster card is convenient and useful for paying for trains, buses and the Underground while in London.
The article talks about taxi, black cab and private car service from Heathrow, but the link for the private car service guides you to a Viator website that talks about rides starting at $92 USD. Those certainly exist, but that is definitely at the high end, closer in cost to an official black cab or taxi into central London. We tend to use the regular train from Gatwick and a car service from Heathrow.
You can find highly recommended car services, such as Blackberry Car (that's who we have used), that offer flat rate services into London for around £30 to £45 from Heathrow. They are very convenient if you are staying at a hotel that is not near the train or tube stations, so you don't have to switch modes several times while hauling luggage along.