Both US and Canada collect tax depending on your residence status. Both tax a resident based on his world income. You pay nonresident income tax to one country if you have bank account interest or investment income in that country and you can then claim foreign tax credit to the country which you pay the residential tax. You have to work with both tax department to determine your residence status. In the year of moving, you are likely to be double taxed. There are forms you should fill out and submit. You will get an answer based on the information you provided. You can then file your tax accordingly. Different people has different tax situation and the answer to such question varies. One situation I know of is that if you are getting benefits from one country, you are likely to be tax by that country as a resident.
Do your own research and call the tax department if you don't understand. There are nice people there.