I've never technically made it to 37 weeks, so here's my experience... This one and the last one both came at 36w6d, spontaneous labor. Both babies were average newborn size (7 lbs 4 oz, 7 lbs 8 oz, respectively). Both were born out of hospital with a midwife (CPM). Both had signs of truly being 37 weeks (plus dating u/s early on and charting confirmed) - they had a lot of vernix, and other signs of being 37 weeks.
First baby had perfect APGARs, but had some difficulty nursing for about the first 5.5 weeks (he had a "short tongue", as best as we could tell by looking up our "symptoms" in the Breastfeeding Answer Book with an LLL leader). I had to pump and spoon feed 1/2 oz before transferring him to the breast for about a day and a half, and then he finally started latching on without the pumping. He was VERY efficient at nursing once he was latched properly. It was getting to the proper latch that was difficult for him. He also did get jaundiced, head to toe, but he was not lethargic, so we just did sunlight and plenty of breastfeeding. He got better on his own and never had to go to the doctor.
Second baby had a little trouble breathing at first, but nothing scary. He just had some grunting and retraction that first few minutes. And he didn't cry right away, but once we got him crying, the breathing issues went away pretty quickly. No O2 was necessary. Baby stayed in my arms (with cord still attached) in the water the whole time, and we basically just rubbed him good and they repeatedly checked his breathing by listening to his lungs and paying attention to the grunting and retracting. After that first few minutes, he was breathing completely normally and was absolutely fine. He got a tiny bit of jaundice, but not as bad as his brother. It only went about halfway down. He's almost completely cleared up. Still see it in his eyes a bit (he'll be 2 weeks old tomorrow). This little guy nurses like a CHAMP. We had some lazy latch issues that caused me some excruciating nipple pain, but the baby still gained 2 oz in the first 3 days of life and is up 1 lb from birth as of 10 days! So he's definitely nursing REALLY well, and we've fixed the lazy latch issue and my pain is pretty much gone. He's a very efficient nurser like his brother, and he also doesn't comfort nurse, which is how his brother was as well. Both of these boys are completely different from my first, who was a 29 weeker and was a big comfort nurser.
Anyway, my experience having a preemie and having 2 "37 weekers" (even though they were both 1 day shy of that
) is that they'll have some preemie-like tendencies most likely, but usually they do fine if they're coming on their own at that point. I flash cook my kids, so I expected mine to come that early. I would have been completely comfortable delivering at 36 weeks, which is when my midwife goes on call. I know how my kids do when they come early... my 29 weeker did really well for his gestation, and both my 37 weekers were developed enough and able to breathe as well as any full term baby. I didn't have any preterm labor or anything like that, but when you do, that actually causes the lungs to develop a little quicker, so PTL babies often do fine coming at 36-37 weeks. A friend of mine has PTL starting around 14-16 weeks, and she delivers them around 36 weeks when she's taken off drugs and bedrest. They've all done well. I think the last one did require a couple days in NICU... she may have come before 36 weeks. I know her first 2 were right around 36-36.5 weeks, and they were both "take home babies".
I would NOT, however, recommend inducing for a non-medical reason (which I know you're not even thinking about that), as that could end up with a NICU baby. But a 37 weeker coming on their own, I'm completely fine with.