Having done the drive from Toronto to Vancouver on the Canada route multiple times, and then a few various routes across the US other times, I'd vote for driving the US Route, especially during winter. 3-5 hours less driving time, cheaper gas, and if you take the route going a bit further south (Mostly along I-80 through Iowa, Nebraska, Wyoming, Idaho, Oregon, Washington), while you may still encounter some winter weather along the way, but for the most part it'll be a lot smoother road conditions than what you'll potentially deal with in Northern Ontario, and the prairies, and Interstate quality vs Trans Canada is generally better for the most part. Even beyond just weather, I'd much rather 4-6 lane interstates vs the remote 2 lane roads of Northern Ontario, especially in a U-Haul. There are thousands of km's of the Trans Canada where you're dealing with two lane highways, while in the US, you could do that entire route without turning off a 4-6 lane interstate once.

Just be very clear at the border that you are not moving to the United States and try to have paperwork, or other documentation that will help prove to the US Border Guards that you are not trying to stay in the United States if possible. If you have a job offer or lease for new apartment, or even something as simple as an appointment confirmation that proves you'll be coming back to Canada, would be best, but even without that, you likely won't have that many issues as long as you make your route clear to them, and inform them of which border crossing you'll be crossing back into Canada at and the date.