I hated the minivan image until we bought one for our daughter. They are actually really neat. Until the minivan we always had full sized vans or SUV/Pickup.
I was amazed whe I had the mini at Home Depot and was able to fold down the seat and easily slide in five sheets of plywood and five sheets of drywall. Just dont buy a tree at the same time.
The minivans are much easier to drive and to park. The SUV is nice because you are up high and can see over other vehicles. SUV's usually have better pickup (acceleration) some of them are rockets. Minivans have a tighter turning radius and hoold a corner/curve better.
WEhen you get into the "pregnant station wagon" Aka crossover, there is less difference between them and a minvan. A Traverse is basically a minivan with less headroom and slightly more passenger room. (and a better engine)
Storage space is about the same (as long as you get the slightly bigger minivan like the Grand Caravan - the regular minvans have almost none). SUVs generally have bigger seats, so they have no more storage space, puls you can stack things higher in the minivan. THe big SUVs have much more space IFF you remove the seats.
The chrysler minivans are really easy to convert to hauling. Often I am out somewhere when the need to haul comes up (i.e. visiting my brother and he says "Do you want this old TV we got a new one). With a minivan at least Dodge, you do not have to remove the seats, they fold flat and/or disappear into a hole in seconds. with SUV's you usually have to plan ahead and rmeove the back seat and leave it in your garage.
SUVs can tow. Minivans cannot, not really.
Very few minivans have 4 wheel drive.
Minivans are much easier to get in an out of, but a big SUV is much more comfortable on a long trip.
You can do ok with two kids in a sedan, but you will frequently wish that you had more space, or you will end up taking two cars. With two kids, you will often be bringing 5 or more kids along with you when you go places. We have 5 kids, and when they were younger, 12 was a typical number for any excursion (so we had to take two large vehichles). Even with just two kids, if you want to take grandma and Grandpa somewhere with you, you need room for six. A few sedans can uncomfortably cram six inside, but it can be pretty unpleasant if you are going any distance.
Minivans and full sized vans have much more room to sleep if you need to sleep inside your vehicle (something that come up with some frequenscy, especially now that the kids are teens. One daughter would arrive at work an hour early to avoid traffic and then just sleep in the car for an hour).
While I like the ideas incorporated into the Dodge minvans much better than any other brand, they come with their own problems. If you buy a used one made before about 2005, set aside $1800 for a transmission rebuild. Lots of other things seem to break as well, but then they are much more plentiful and far cheaper than other brands. If you buy a new one, they suposedly solved the transmission problem, but we looked at the new ones, and they look really chintzy to us.
By the way, The Dodge Durango and the Traverse and its sister vehicles are almost the same vehicle. They are only a few dollars differnt in price. The Traverse has a few more storage pockets and get 1 mpg better mileage. However the Durango offers a 390 HP 5.7 litre Hemi. If you want a SUV rocket, that may be the better choice.
All of the 6 plus passenger vehicles get about 15 to 20 MPG. I dod not see that it makes any difference which one you get in that regard.
For your size of family, you could also consider a pickup. You cna easily haul four or five people in a four door pickup and have plenty of room for stuff and dogs in the back. Plus if you want to buy a tree, or some drywall or a big tool, you are all set.
I have never felt that the type of car that I drive is likely to provide any signficant advantage for safety (unless it is 4 wheel drive). If you are in a head on collision, or hit by a train, you are probably going to die regardless of what type of car you are in. It is very very rare that an accident severe enough to kill you will not kill you if you have a different type of car. It is more important to drive carefully, stay off the phone, and wear your seatbelts, than it is to drive a particular type of car. I know one mother who always picks cars by safety ratings to "protect her children". She chastises us for driving such unsafe vehicles as a full sized conversion van or giving our daughter a used Dodge Grand Caravan. Then she drives around with her kids talking on the cell phone and even texting.