Welcome to the RoadTrip America Forum!
The two routes you have linked are relatively straightforward; of course, the Northern route being shorter than the Southern route by the good portion of a day.
Both of the routes you link to could have weather detrimental to your trip. The difficulty of weather is that it is impossible to tell what it will be like more than a few days ahead of time. Just this morning I was reading of bad weather and traffic problems along I-40 near Amarillo, TX, but that's not any indication of what to expect a month from now.
I would look at more options than those you have listed - you could run on I-40 3/4 of the way across the country and then hit bad weather, so you're going to want to have an alternate route available to you should that be the case. Likewise for I-80/I-90. For example, if you are heading across I-40 and bad weather is going to hit Arkansas, you could avoid it by going up I-44.
I would also recommend, if it is possible, to have somebody else along with you on the trip, to help out in case a problem does arise. Normally I would be all for recommending taking a solo trip, but I think this particular situation calls for a bit more caution. A cross-country move can be stressful enough.
I see that you have added in a stop in Kentucky on this trip. Any particular reason?
Finally, allow as much time as possible to make the journey. It will take five days to cross the country under normal circumstances by the quickest route. With the way the weather has been thus far this year, I would definitely allow more time in case you have to stay put for a couple of days.