A quick note: much of the the east coast of the US in the summer is HOT. Hot during the day, hot at night. Sleeping in a tent is going to quickly become miserable and sticky, and you’ll need to contend with bugs as well. I am a tent camper, and I do not do so in the summer. Sleeping in an SUV will be equally hot and miserable, but not as buggy.
Rent something with an a/c you can use.
One option would be to tent camp, then ship the gear home. You could use it again in the future, or sell it from a more relaxed and advantageous position. Or, since I enjoy tent camping, you could ship it to me. 
If you tent camp, the right gear can make all the difference between enjoyment and misery. The two main things to consider are a tent that WILL NOT LEAK (i.e., not the cheapos) and sleeping pads or an mattress that will provide good comfort. I’d suggest consult with experienced campers for advice on these.
To have A/C while camping, you’d need a camper with an AC (electricity) powered unit and a campsite with electricity. The A/C unit would almost certainly be mounted on the roof of the camper, which could be problematic.
The alternative is getting a breeze. This shouldn’t be hard with the right tent, but would only be possible in a camper if it had screens – otherwise, you’d probably get eaten alive by bugs. You’d have to select a tent carefully to have both significant ventilation and significant storm protection, as those tend to be mutually exclusive design features.
Whichever way you go, keep an eye out for state (and sometimes county and city) parks as well as commercial campgrounds. State parks tend to have lower fees and more “out in nature” surroundings. Commercial campgrounds tend to have more amenities (e.g. convenience stores) and locations closer to cities.
The baby Pedants grew up camping until one day they (all girls) figured out what a luxury hotel was; after that “camping” meant Holiday Inn instead of the JW Marriott…
I don’t think your question is easily answered. Vehicle-based tent camping is still a lot of stuff to carry around, and if you move often it is a lot of packing and unpacking. It is not a great way to spend weeks unless you are very experienced, and you find campgrounds with hot showers. Motor-home camping is more organized but more vehicle to lug around. It can be a fairly expensive vehicle rental. Good campgrounds can be surprisingly difficult to book during peak season.
If you know your destination in advance, Hotwire and Priceline can be your friends over any guidebook hotel prices. You can always take basic tent-camping stuff and just use it when you cannot find inexpensive accommodations.
Most likely you’ll suffer a bit no matter what you choose and at the end of the trip feel like it was the most fun you ever had. That’s camping.
One option worth looking into: cabins at the state parks or KOAs. Some of the state parks out west offer cabins and yurts, which are pretty nice. KOAs have “Kamping Kabins”, which are sometimes decent, but sometimes not so great. The prices aren’t as cheap as you’d think, but if you want a camping-type experience, they might be what you’re looking for.
Renting RV’s (camper van) is not cheap. You could much more easily find cheap motels along the way, that would be less expensive. Most people rent RV’s when traveling in the western US, where the population (and hotels) is less dense and the scenery is much more so.
Good luck.
I would camp during the week and stay in hotels on the weekends (Fri & Sat nights). Campgrounds are going to be packed on the weekends but will open up when the weekenders go home. You’ll be able to refresh yourself during the weekend with guaranteed hot showers, a soft bed (hopefully), and maybe even a breakfast you won’t have to cook yourself. You’ll be ready to camp again on Sunday and camping is WAY more enjoyable when the campground isn’t full.
Rain can put a damper on camping, but you can always use that time to catch up on reading, or see some attraction on your route that’s indoors. If it storms you can always check into a hotel.
I’m not sure if I would go the SUV route. You’ll pay a lot more for the rental and for gas. And while you could sleep in it, it’s not going to be as roomy as a tent (assuming you invest in something large), the ventilation won’t be as good and bugs can get in with the windows open. Plus, unless it’s a super huge SUV, you’ll have to shuffle your stuff around to make room for your sleeping area each night. With tent camping you can keep the car packed and just pull out the stuff you need each night.
ETA: you can buy small fans that attach to the side or roof of your tent. Couple of those and a nice cross breeze and heat shouldn’t be an issue.
If you want to tent, you may be able to rent it. Where I am both the local University (as part of the thier outdoor recreation program) and the local outdoorsy store rent camping equipment.
Brian
But rarely for a month. At nearly any reasonable rate, you’re still better off buying for that length of time.
We live in Seoul. Both of us have camping experience - we’re not complete newbies. We just don’t go camping in Korea (not a lot of good campgrounds). And we really don’t want to drag excess luggage back and forth. Even if we brought the stuff back with us to Korea God knows when we’d use it again.
We’d like to see as much as we can while we’re there, which is why the trip will be one way.
Anyway, thanks for the advice everyone. 
Since you’re coming from Seoul, I’d just stop at the first Target or Walmart you can find in Atlanta to buy your equipment and then just dump it at a Goodwill store before you fly home. Consider it one of the expenses of the trip. The amount you’ll pay for the equipment should be no more then the cost of a few nights in a hotel. Both Target and Walmart have pretty good selection of equipment, but nothing fancy or specialized like you’ll find at REI (but not nearly as expensive, either).
Or, if you’ll have a laptop along, you could try advertising on CraigsList for your departure city maybe a week after you start and refresh it every few days. I’d make the deal as simple as possible: one price for the whole lot. If someone is interested, tell them to be at whatever meeting point on this date and time or they miss out. And then be ready to haul it to Goodwill if the buyer flakes.
East Coast…its cheaper to rent a car an get motels off the interstates.
KOA cost as much as a motel room and you get a bathroom w/Hot water and AC n Heat.
I agree with those who recommend renting a full-sized SUV and sleeping in the back. It’s the perfect balance between tent and camper, and it will allow you to navigate city streets and traffic. When I drove from Georgia to Alaska, we had a memory foam mattress topper in the back of the Chevy Tahoe and slept back there at least 3 or 4 times during the trip. We stayed in hotels for the vast majority of the trip. But anyone considering camping in a tent to be an option would be extremely comfortable sleeping in an SUV for the entire month. There’s zero set up time; just pull over and sleep. You don’t have to pitch the tent, repack the tent, fight mosquitoes and horrible weather… you can run the A/C or heat as needed… you’ll love it.
Pretty safe bet this trip was completed about ten years ago. Further suggestions not likely to help the OP much. 
Tricky Bear, the necromancer!
I read the title and thought, “Camper Van (Beethoven), I know them. But I’ve never heard Tent.” After more time than I care to admit I realized this isn’t in CS.
Nothing Camper Van Beethoven did could possibly equal Tent.
I’m gonna, I’m gonna, I’m gonna, I’m gonna, I’m gonna
I’m gonna, I’m gonna get you in my tent, tent, tent, tent, tent
Wo wo, it’s only common sense sense sense
I know that you won’t mind the stench, stench, stench
Of the sacrament
Every time I see the thread title in the forum, my brain keeps reading it as “Camper Van Beethoven,” too! 
Tent: sleep on ground or hammock.
Van or camper: sleep above ground.
Decade-old zombie thread: sleep it off!
We’ve had no trouble taking our 25-foot-long 7.5-foot-wide class-C Sprinter-platform RV along odd San Francisco streets and I don’t fear Manhattan. Take scooters, bikes, or taxis in crowded places, then return to king bed, good bath, clean kitchen, comfy couch, and WiFi. We’ve hooked into Elks’ Lodges in and around major metro areas; Elks are friendlier and cheaper than commercial ‘camps’ but we’ll take whatever we must.
We tent-camped exclusively until 25 years ago and we’ve tented some since but damn, tenting in and around cities can get dicey! Don’t be mistaken for homeless.
I would instead just go online and use the cheapest hotels you can find or airbnb and sorts. Some people just rent out a room in their house. This way your just driving the car. You will appreciate having a place to shower at night. You can have a small 3-4 person tent and some air mattresses which should fit easily in your trunk and maybe on occasion sleep in that if your going thru a camping area.
For food we just ate out of a cooler.
If your not driving thru during a popular tourist time rates shouldnt be that bad.
Just to note for recent posters to the thread: as GaryT noted, the OP was made nearly 10 years ago, and HazelNutCoffee hasn’t been active here in two years.
Wow. Okay.
There’s no way that I randomly resurrected this thing. There must have been a deleted spam post or something?? Strange.