Let me first point out that there are two other "major" testing companies you haven't mentioned: MyHeritage and Living DNA.
There are plenty of people who reflexively say "Ancestry DNA." But that's not right -- the best company will depend on your own goals and resources.
If you can answer some of these questions, it will be easier to help you choose:
* do you want a sample stored for possible testing years into the future? (say the person being tested is elderly and might not be around to do additional samples if desired -- this happened for one of my close family members and I am so happy I have a stored sample)
* do you want the largest possible pool of matches?
* Where in the world do you expect most of the matches you are most interested live?
* do you want a company that provides the tools necessary to identify your most distant cousins (i.e., deepest in your tree)? Keep in mind that even with these tools, this takes a lot of learning and work?
* do you want to trace deeply your paternal line (i.e. dad's dad's dad's dad's ....) or maternal line (i.e., mom's mom's mom's, ...), more deeply than a standard DNA test will allow?
* are you concerned about these companies' sharing your data, usually for "scientific research"? I haven't read all of their ToS's, but if this is a big concern then you probably should read one before testing
* are you concerned about the new use of these matching services to identify perpetrators of heinous crimes? Detectives are now uploading crime scene DNA to these companies to look at matches and try to identify perpetrators. Some of these companies have moved towards allowing this, and others make it very hard for police to use in this way.
* how much are you willing to invest in testing? Are you willing to upgrade or do new tests on an ongoing basis, or to pay some money every year to access all of the features of the company where you tested? Or do you plan to do one basic test and be done with it and you want to keep that cost minimal?
* do you want the option for health information?
* do you want decent ethnicity matching? which ethnicities do you expect? (Do you have a particular interest in the British Isles?)
There are probably other considerations, but these are a good start.
A final point -- the health information and ethnicity information is just for "entertainment" value. Except in rare cases, it doesn't provide direct help for family tree research. The health information should not be used by itself for medical diagnoses.