Since the US Army was so small between the Mexican War and Civil War, the generals will be those who graduated from West Point (fact--75% of US Army officers in 1860 were West Point graduates) or other military schools--and there weren't really a ton of them hanging around in 1861. It's just without a Mexican War, there wouldn't be as many indications as to who is particularly good and who isn't.
I think for a lot of the Napoleon-obsessed West Point officers, no Mexican War might be helpful for the Civil War experience, since hopefully it would catch on faster that Napoleonic tactics and 1860s warfare are an ill fit. The earlier years of the war had tons of mistakes like that.
Overall, it will affect the way the war will be fought, and possibly lead to less casualties although its bound to be bloody either way.
We would almost certainly have a very different set of generals and with very different experience. Many of the ACW IOTL might have instead left the Army (Grant, Lee, Sherman, Meade, etc.) and might not have come back or, if so, might have come back far more junior, and less experienced, than they were IOTL.
So, whether we have the same or different generals, their 'experience' will largely be limited to what they were taught at West Point and perhaps some garrison duty or 'Indian Fighting.'
However, I don't think that fact is going to make them realize the change in warfare that technology has wrought, nor make the war any less bloody.
I suspect that the early years of the war would be even more error prone (on both sides). Either side could come out having an early advantage in generalship. If it's the North, then the CSA is crushed faster.
One other note, the lack of the Mexican-American War is also certainly going to have an impact on the military equipment available as the war did provide a good test for lots of equipment and no doubt contributed to some further development or acquisition. So, both sides might start with less capable rifles, artillery, etc.