60 parts makes it easy to divide by 2, 3, 4, 5, 6, 12, 15, 20, and 30.
As a competitive advantage, people can work with angles, times, (as they still do) and currency, distance, etc. in early civilization where large complex buildings, complex finance, and dedicated investigation of math & scientific problems are first being made.
We don't have much trouble with decimal currency, and rounding off or approximating is not as much of an issue as with building something where you want the corners to line up and whatnot.
In order to easilyndo these things, they might have developed a head for rational arithmetic earlier, or even decimal fractions and long division.
What would change would be some other way of manipulating quantities and values that practitioners can handle with ease and efficiency. Even if you had the ability to work with fractions in clay tablets, how do you show that on your ruler? I'm thinking the instruments mightnbe pre-marked with fractional values that can be constructed geometricly, when making the primary markings. So, even though the ruker or compass is marked 1..10 units, there will be marks at 1/3 of its length, 1/4, etc. Then you need the same fractions marked with a denominator of 9, etc. You end up with a maze of marks... or do you? Systematically you end up with each unit divided into 12 subunits.
This is exactly how the modern Westwrn scale in music comes about. Start with the simple ratios and fill in all you can reach using what you made thus far.
So why didn't they invent it anyway, for professional tools even if not in popular use? Maybe the culture didn't care for these simple ratios, cared about others like 1/7, and was drawn to non liner-repeating patterns like geometric progressions, golden ratio, and other things that produce irrational values. Now, having easy divisions doesn't help much since those are all ugly unused values.
So my answer is that art and architecture would reflect these sensibilities instead, like fractals and immitations of nature. That would influence early scientific inquiries and they would use geometric progression, essentially logarithmic scales, rather than 60ths linear scales. Maybe they would discover $ e$ early on, and rulers would be based on vanishing points, and compasses would be marked with the sine of the angle rather than 60 degrees. For finance, they would understand compound interest intuitively.