Hi Gabriel,
There are a number of differences between a treadmill and running outside. Some of them matter and some don't. But first you would have to know if the treadmill is calibrated correctly so that the speed and distance are actually accurate. If you are at the gym and one treadmill says you are consistently farther/faster than another, there are inconsistencies one way or another and actually both may be wrong.
That said, let's look at what is different. To begin with you are indoors and don't have to cope with things like wind or weather. A side effect of being indoors though is that it also simulates humidity. If you haven't already noticed, it probably "feels" like you sweat more, but in actuality it is simply not evaporating as well as outdoors. In that case the treadmill might be more difficult. This you can fix by parking a big fan in front of you, which basically amounts to more comfort. Although since I also mentioned wind, the fan does not solve that, since you are not moving through a mass of air, but a small amount is going past you... if that makes sense. You may also have an issue with drier air indoors, and especially if there is air conditioning. In that case, you might find outdoors more pleasant/comfortable unless there is a lot of pollution . These things may not be a big deal.
Probably the biggest issue, tread mills have a lot of give, so you are generally going to experience less impact than on the roads, and you will have substantially less impact if you are hanging on to the rails at all, because then you have transferred a lot of weight to the rails through your arms (like an instant weight loss program). This is important, because impact over time leads to fatigue and especially in your quadriceps. Quadricep fatigue can lead to tremendous slowing. So generally, you would develop more quad strength running outdoors and be better prepared for an outdoor run. Or to put it simply, your body is most comfortable doing what it is used to doing. Going from treadmill to outdoors is a change that your body may not immediately adapt to.
Additionally, (kinda like what your wife says) treadmills tend to force a consistent pace and stride length... or you fall on your face or scoot off the back. This is a great thing for efficient running, but you don't have the machine helping you outdoors. And, that old nemesis fatigue might set in again... which often leads to a degradation of form and efficiency, and of course when that happens you slow down. Add to that other ineffciencies like uneven terrain and turns that can chop up strides etc and tax peripheral muscles and it is a little different ballgame.
Finally, if you have a distraction on your treadmill like music or TV, you have generally turned your brain off to the running, which can be quite a good thing. The more you worry or think about your running and/or pain, the less you are likely to perform well. Just compete and things go well. It could be difficult to find that level of distraction outdoors when you might be worried about things like jostling elbows, cars or dogs for example.
Having said all that, there is a correlation. If you start going faster or farther on a treadmill, the same should be true outdoors. And running faster... indoors or out is something that does benefit your overall running. But, measure like against like. Measure the treadmill against the treadmill and outdoors against outdoors. You wouldn't try to measure an uphill run against a downhill run. But ideally, you would have a mix of types of runs, distances and paces for overall conditioning. And if you have a specific target, like a flat, hilly, dirt, asphalt or otherwise 5k, you would want a training program that approximated those conditions in it's preparations. Or like I said earlier, your body is most comfortable doing what it is used to doing.
So my answer? You certainly could have days where everything lined up and you ran as fast or faster on the roads than the treadmill. But generally I think you would find your treadmill times and efforts to be less work than comparable outdoor running.
