The answer is not as simple as you would like it to be.
Firstly, they are not roughly the same size. A .45 caliber is quite a bit bigger around than a 9mm. However, the diameter of a bullet is only one component of the overall ballistics characteristics. Other components include the weight of the bullet, the bullet configuration, the powder charge used, and the length of the barrel from which it is fired.
Let’s talk weight first. In general the .45 cal is a bigger bullet and thus is heaver. This generally means a projectile that will loose its energy (slow down) quicker. However, being heavier also means more ft/lbs of energy delivered at the impact site than a lighter projectile traveling at the same speed.
Ok, now lets talk bullet configuration. There are many different ways a bullet can be put together ranging from a hollow point, to a fragmentary round like a “glaser” to a fully jacketed “hard ball”. All of these different configurations behave differently upon impact. A hollow point round is designed to mushroom, which causes it to loose all of it’s energy relatively quickly on impact. The result is that shortly after penetration all of those ft/lbs of energy are more rapidly dissipated into the target. Contrast that with a fully jacketed round that is designed to hold its shape better and will simply poke a nice clean hole.
Next you must consider power charge. It takes a bigger charge to fling a bigger bullet. The .45 has a larger diameter so can accommodate more powder. However, it is easier to fling a smaller bullet like a .9mm faster. There is also bullet weight to consider. Just because a bullet is bigger around does not mean that it is heaver. So, it is a balancing act between charge, diameter, and bullet weight regardless of what caliber you are using.