1. Without question it will be easier from HR Hakone. When you said you might have 4 nights I would have recommended 2 at Odawara and 2 at Hakone; with only 3 nights and definitely 1 night Goran, I would not recommend Odawara logistically.
2. Both the Hilton and the Hyatt have onsen as in a bath fed by a natural hot spring, but the Hilton onsen is more representative of a Japanese onsen, while the Hyatt is just a large bath that you would not know was fed by a natural hot spring unless they told you. That said, the bath at the Hyatt is more luxurious and is sometimes very sparsely used, giving it a private feel. The Hilton onsen has an open-air bath, but no view.
3. The Hilton would comp upgrade you to a Deluxe room, or possibly a 4-bed Deluxe room (both 54 m2), but not to the Japanese suites or a cottage; the cottages look interesting but the short hike to the main building, such as to go to the onsen, could be a pain, and they have no or limited view.
4. The public areas of the Hyatt are much more luxurious that the Hilton, but I did not think the quality of the finishings in the room lived up to the photos. The Hilton feels like a hotel, the Hyatt feels like a country retreat. The Hilton has spacious grounds, lots of resort facilities and activities, outdoor jacuzzis, pools, etc. The Hyatt has comfortable living room with a nice fireplace, but nothing else by way of grounds or facilities. The Hilton has plenty to keep your kids busy; they'll be bored if they have to stay all day at the Hyatt. But with only 2 nights, you'll have plenty to do sightseeing on the ropeway to Lake Ashi and Owakudani, so the only problem would be if it rained the whole time you are there.

All that considered, I think the Hyatt wins hands down.

We personally had some issues with the Hyatt, combined with what we perceived as an air of superiority. But no property is the perfect experience for 100% of the guests 100% of the time. So in spite of our own personal preference for the Hilton, my recommendation would be the Hyatt.