You can't necessarily trust the displayed pace on a treadmill, for a number of reasons:
1) Poor calibration. Very few treadmills are perfectly accurate, and they can get worse over time. You can use a device like an NPE treadmill sensor to measure the exact speed that the belt is moving, but even that isn't perfect because...
2) Treadmill belts don't move at a constant rate. When your foot is planted, they slow down because you've just dropped a heavy load on them. When you're airborne, they speed up. Even if a belt is averaging, say, 8 miles/hour, it's going slowest when you're actually on it. The difference depends on the quality of the treadmill. Better treadmills with more powerful motors, bigger rollers, and better speed control boards will eliminate more (but never all) of this difference.
3) Treadmills may not be perfectly level. Even at a nominal 0% grade, you might be going slightly uphill (either because of the treadmill itself or the floor it's on). It may not be a ton, but it could easily be worth a few seconds/mile.
Assuming you have a calibrated, flat treadmill going exactly the same pace as outside, there are still some differences. A few things can make the treadmill feel harder:
1) Heat buildup. I run on a treadmill in my garage, and in the winter, it can be 40 degrees in there. I am typically shirtless after the first two miles, and very comfortable, even though I tend to run cold when outside. You will very quickly build up a zone of hot air around you and cook. A fan makes a huge difference.
2) Lack of familiarity (physiological differences). If you aren't used to running on a treadmill, you might be running with weird form, stressing about falling off, etc. And it's not like the differences in your form would be easily detectable by the naked eye. Only a slight increase in muscle tension can make running a lot more difficult. Until you can just comfortably cruise on a treadmill while letting your mind wander, you aren't really used to it yet. You probably are working harder on a treadmill until you've put in a few hundred miles at least.
3) Lack of familiarity (psychological). If you're just starring at the numbers and you feel like you're on some kind of torture device, it's going to feel hard. The time can crawl on a treadmill. I find that the more you do it, the easier it gets, but it's worth listening to some good music, old race videos, or whatever keeps you pumped. At least for hard workouts. For easy runs, I just zone out and watch dumb TV or listen to audiobooks.
That said, if you're used to running on the treadmill and you have a cool environment, it should be easier than running outside.
1) The lack of wind resistance is real, and most fans aren't going to replicate it. At hard workout paces, this is easily worth 5 seconds/mile.
2) Any non-research-grade treadmill is going to have a lot of bounce. This allows you to offload some of the metabolically costly cushioning that your muscles would have to do onto the deck of the machine. This is basically what super shoes do, but it's even better because you don't have to drag around the weight of the cushioning on the end of your feet.
3) Perfect pacing makes you faster. Even if you're doing mile repeats just following the shoulders of some guy who's a human metronome, he's not running as evenly paced as a treadmill. You will always be faster running evenly, and you will never run as evenly as on a treadmill.
All of this together makes a huge difference. I do a lot of treadmill training, and I can now run a half marathon about 3 minutes faster on a treadmill than I can on the roads.