Say you can do maximum 10 pushups. Supposedly you should go close to failure to have a good training effect. But the point is to break your muscles down down, to rebuild them stronger. Is there a big difference between 9 or 10 reps?
It's recommended not to go to failure because it's demanding on the central nervous system. If you just do some pushups from time to time you can go for failure no problem, but if you have a routine with lots of exercises and sets per exercises it can get too much.
Generally the point of an exercise is to push your body to adapt to stress placed on it. You want each workout to push your body a little bit more than last time so your body gradually adapts towards where you want it to go.
How you push it is down to your goals.
If you want to be able to do lots of push ups, you will want to always try to do more push ups than the last time. That is the difference between 9 and 10 pushups. If you did 9 pushups last time, this time you want to do 10. Next time do 11.
The difference is huge! Mentally you are pushing your self and motivating your body to push itself. Physically you are pushing your self and your body to adapt to doing more pushups. If you always do the same number of pushups every time you will just get stuck in a rut and go nowhere.
It doesn't matter if you go to failure or not. You just need to do a bit more than last time. Sometimes this will take you to failure, sometimes not.
The simple answer is 1 rep - not a big difference. What I would recommend instead of adding volume is to add change or resistance to push yourself physically and mentally. Simple variations, such as: single hand, clapping pushups, hands wide, hands narrow add variation and ensures that your body will not 'adapt' to your workout and that your using muscles in different ways. You can also change from 3 (or 5) sets of 10 to 1 set of 50.....the idea is to push yourself and improve with each workout.