You will train harder
You are much less likely to put in a ‘half hearted’ effort at the gym, if you have to actually write it down and have those numbers in black and white as a constant reminder of how you didn’t put in the effort that day.
It will keep you focussed
There are some amazing apps for logging your progress, and they have a place. Just not in the gym! Does your phone go to sleep while you are doing a set? Fiddling about unlocking it is just an extra step you don’t need.
What if you get a text? Sure you don’t have to read it, but it will be in the back of your mind, wondering how important it is. Distracting.
Simply write it into your gym log book, then when you are having your post workout shake you can update your app with the information from the diary.