If you're leaving DCs alone, then for a roll where you need a natural 4-7 to succeed, this system basically grants a +2. For a roll where you need a natural 15-18 to succeed, this system grants a -2. I don't know how noticeable that would be, since you're succeeding/failing one time in 10 when you wouldn't have otherwise.
Another relevant comparison is how often the creature with the better bonus winds up succeeding at a contested check. Since one of the main goals of this change is to reduce the likelihood that the creature with a lower bonus wins the contest by sheer luck, an important evaluation of this change is how much difference it makes on this measure.
Here's the comparison between this method, the ordinary 1d20, and the 2d10 method. Your method falls between the other two, but is closer to 2d10 than it is to 1d20. Certainly an improvement over RAW, IMO.