Pro Tools or Logic? (or Cubase or Sonar?)
Let me respond to your question with a question. Your favorite website? Does it run on a Windows or Linux server?
You don't care what kind of server it runs on, do you? Or even if it's a Macintosh or Unix server, it just doesn't make any practical difference to your user experience.
So in some pro audio applications, it doesn't matter which DAW you use as long as it suits you.
The 'big four' DAWs are all very capable of turning out work of professional quality, so if you are supplying work to a client, then the quality of the work is all that matters, not how you made it.
But there are two areas where it does make a difference...
Do you want a job in audio, rather than a freelance career? Then you should learn Pro Tools. Not only is this the most commonly found software in the professional environment, it is also the most generic in the sense that skills learned transfer easily to other DAWs if necessary.
Rightly or wrongly, Cubase - to pick an example - has developed a reputation as a 'bedroom studio' software and on a professional CV or resume it may fail to impress.
The other place where you need to be using the right DAW is if you operate a studio for hire. When we go to Abbey Road for example, we record using Pro Tools. You can't sensibly operate a pro studio for hire without centering on Pro Tools, although you could offer other DAWs as options.
In the end, all of these softwares have the same functions to fulfill and it is far better to gain skills with the fundamentals of audio than worry about which DAW is best.
By the way, there are other DAWs besides the 'big four' we have picked out. Your choice of DAW will be very personal, but to us nothing matters apart from the quality of your work!