How does Jordan compare to Egypt?

The hassle factor is very much lower than in Egypt!

If you survived Egypt you'll only laugh at the ocassional tout in Jordan. They're far less professional than Egyptians.

According to my own personal experience, hassle is not lower. It's just zero. Ok, I only stayed 8 days in the country, but visited all the major tourist places and only once in Petra I found a little girl insisting that I should give her a small stuffed animal hanging from my backpack. Well, 'insisting' is probably not the right word though. She asked like three times to give it to her, I said no, and that was all.
If you love nature, do consider a trip at Dana Nature Reserve. It was the highlight of my trip!
In Aqaba you can go on a private boat tour and ask the boat guy to take you to a place where you can snorkel. The day my group went, however, it was too windy and nobody dared to take off one's clothes. The boat tour was nice though.
Happy travel!

I stayed 7 days in 2003 and I was offered donkey/camel/horse rides, souvenirs, etc. in Petra many times.

They tried to overcharge me several times in Wadi Musa for bottled water.

Tried to overcharge me in a bus from Jarash to Amman.

It's not nearly as bad as in Egypt, but it exists.

scythian, this is not meant to contradict you but I guess I just have a higher threshold of tolerance due to the fact I'm...accustomed. In some parts of my country (especially big cities and the south) the situations you describe occur daily (well apart from the offer of a camel/donkey ride!) and I imagine I'm just so used that I regard it no longer as hassle (if I have ever considered it hassle). Just the other day there were news that a couple of Japanese tourists were charged over 500 euros for a meal in Rome... so you can see why overcharges for bottles of water are the least of my worries :)
(Sorry PinkNosedPenguin, this is a bit off-topic)

For those who love nature Jordan is a paradise and I think your children will love the adventures that Jordan offers.
In Dana as sushi 75 suggested they can explore the ancient village of Dana, and join nice hiking in ecological reserve. In Wadi Mujib they can explore the Canyon which has several waterfalls , natural pools and also the Dead Sea. I have not seen animals there just a snake and some fishes may be that at dries trails you can see animals- the Mujib reserve have a trail wich name is Ibex trail .

Very nice is also the Azraq Wetland Reserve where can see many birds and also buffalos. And at Wildlife Reserve Shaumari they can see Gazelle, Oryx, Onaga, Ostrich, Cheetah, hyena and wolf. In all these Reserves are offered various programs for hiking and wildlife observation. They also provide accommodation in hostels and campings . The website to you see the Reserves and programs : http://www.rscn.org.jo/orgsite/wj/Group2/WildPlacesOverview/tabid/170/default.aspx

Another nice program for Kids is the visit to the desert castles, the castle of Ajloun and Jerash.

Almost zero harrassment but also a lot less general friendliness too. Its okay but they just don't care too much, unlike in Egypt. It is also the only ME country where I have encountered downright hostility, although not too bad.

The place where there is some harrassment is at Petra, especially when they see you with kids! First is the horse and cart for a breakneck run down the siq, for an outrageous $40 or more. It must take all of 10 minutes too. So I would avoid that, walking through the siq slowly is wonderful. Then the young boys inside Petra with the donkeys will press you constantly to take a donkey around. Can't remember the cost but it wasn't too much, my kids were 4 and 7 at the time so it was quite useful.
Dead sea is wonderful, if you go to the public beach go early before the crowds and before it gets too hot.
Wadi Rum also not to be missed. If you take a four wheel drive tour around it, make sure it is one with an open back, so you can sit out in the fresh air as you go around, better than couped up inside the vehicle for the day, especially in a place as beautiful and surreal as WR.

Very little harrassment and friendly people, but as is the case anywhere in the world, prices go up where there are a lot of tourists. So down at the Petra gate things are expensive and you may get ripped off. Simple case of supply and demand - people seem prepared to pay absurd prices for a 5 minute cart ride down through the siq, so the prices stay up. We saw many tourists riding those carts, but when I found out the price (and fell to the ground when I heard it) and twice tried to negotiate them down they weren't interested. Obviously there are plenty of people who will pay.

We got overcharged on a serveece going from Amman to Madaba, but it was pretty cheap anyway so it wasn't worth an argument.

I can warmly recommend you to stay overnight in a camp in Wadi Rum at a campfire and sleep under the stars. I did it with some kids that age last year. We also had a camel ride arranged there (must be done beforehand.) Climbing the sand dunes and rock bridges in Wadi Rum is fun.
Petra is also fine for exploring, and so is al-Siq al-Barid
Vaniacroccia recommended Azraq Wetland Reserve. I don,t. There were very few birds to observe even if it was supposed to be the migration season when I was there.
Bathing in the Dead Sea is really fun. Bring googles to protect the eyes.
Have fun.

I travelled through Jordan and Egypt in one trip and found that I was hassled incredibly less in Jordan. I started in Jordan, went to Egypt, then returned to Amman to fly home (not great planning but that's the way it worked out). Personally, I found it was a relief to return to Jordan after my time spent in Egypt as I found people a lot friendlier and more hospitable in Jordan.

The part of Egypt that I would rave about is the Sinai peninsula (this is the only exception for me in terms of not feeling I was being hassled). I found the snorkelling fantastic there. Aqaba was a disappointment for snorkelling and didn't spend much time there, preferring to spend more time at Dead Sea, Petra, Madaba, etc.