How is Smoky mountain NP compared to Vermont/NH?
We visited NH in summer 2020 and SMNP in summer 2021.
Smoky mountain range is far more vast than Presidential mountain range. Personally, we didn't appreciate the 'country bumpkin' style tourist traps around Gatlinburg, Pigeon Forge and Sevierville. It was also very crowded in this area. You will get enough of the southerny country feel wherever you go down there, and I quite like it. The touristy version of it though is in your face and inauthentic.
The highlight of our trip was the 3-4 day breathtaking drive we took down the blue ridge pkwy starting in Shenandoah NP. The most impressive parts are in North Carolina past Boone, so skip to there if you don't have the time for the full drive. The views in this area are more impressive than Smoky Mountain NP IMO and you'll mostly avoid the dense crowds. There are also endless opportunities for sights to explore in that area off the pkwy as well as you pass through quaint little mountain towns.
We spent an unplanned shabbos near Chabad in Knoxville, and it was a beautiful experience.
If you enjoy a lot of scenic driving or wilderness hikes, SM is probably the choice.
You will not get this kind of endless mountain wilderness in NH, the entire mountainous area is about a 1:30 minute drive across, but it does have a certain open feel you wont get in SM. The presidential mountains seem to be more spaced out and not clustered together as closely, giving it a more vast and open expanse kind of feeling as you view them. In this way they are more reminiscent of a (very toned down) western rocky mountain range. Driving up Mt Washington is definitely a unique experience. There's also kosher food and minyanim in NH.
If you enjoy parking yourself in one spot while on vacation, you should probably choose NH.
If you choose NH and have extra time, coming from NY you can take a detour and add the Lake Placid area and/or parts of the Green Mountains in VT to your itinerary. There's a ferry across Lake Champlain you can take with your vehicle from Essex, NY to Charlotte, VT. The Adirondack area requires some serious hiking to get the views though.