I am suppose to explain how and why each country chose a different way of establing these systems. Any ideas why they chose different ways?

In what way are they different?
Transcontinental, they both cross the continent.
Basically the same.

Geography, plain & simple.

Each transcontinental route had to conquer the Rockies & the Sierra Nevada Mountains, in the Middle US, the Rockies & the Cascades for the northern US routes, & the Northern Rockies & lesser ranges (but no less formidable) across the Canadian expanses.

The routes followed the way of the lesser grade, that grade dictated by the natural obstacles. Even so, all these transcontinental crossings posed monumental obstacles that had to be overcome. With the 1st Transcontinental Railroad (US), it was the Engineering, courtesy Theodore Judah for the Central Pacific & Grenville Dodge for the UP, that conquered these granite fortresses.

The southernmost routes, like the ''Sunset,'' had their own obstacles to be overcome whilst trying to traverse the deserts of the southwest, as well as across the gulf region, where swampy conditions posed their own unique problems.

When surveying, constructing & running any railroad, grade is the primary concern, since battling tonnage is hardest where gravity plays most the spoiler. Always has been, always will be.

Wow, l really have no knowledge of how the Canadian system was established.
The U.S. was built by private enterprise, it was the only way to settle vast sections of the country, getting people & goods away from the established waterways.
So congress authorized ''land grant'' railroads, as an incentive to get private indsutry willing to spend the huge amounts of money required to build the railroads.
For every mile of track built they were given tracts of land in the areas they went through.
Of course this type of system encouraged huge amounts of fraud & outright theft but overall the country benefited by having the lines built. For instance the tracks were not always built in the most direct route, the railroads purposly made detours in some cases so as to acquire more land.
The U.S. government could not afford to build the lines themselves, with this incentive, the private companies could be do it.
As repayment, all land grant railroads were required to move all goods & personell for the government for free until the debt was considered setteld which was after world War l at which time they were considered settled.

l am an engineer, but l do not know a whole lot about the Canadian Trans R R. l think Rango is on to something here though. The U.S. ''capitalism system'' allows private businesses (like railroads) to compete with each other. This keeps consumer costs low because businesses have to compete with each other to keep customers. And the business with the best service & prices has the most customers. However, l believe the first Canadian Trans Railroad was funded by the Canadian government only. The government set the transportation prices & there was no competition. Which means the prices would be higher. l do not know for sure the reason Canada did this. Unless that country does not embrace the ''Free Enterprise'' system as wholly as the U.S. does. There is some speculation in this response here. I'd like to be more definitive, & tell u exactly what u need to know, but l can not be positive.

The two links below will give you a history of Canadian railroad building in the early years of the country.
One likely reason for the government involvement in the building of the transcontinental railroad here was that BC was promised a link to a transcontinental railroad within ten years of the province is joining Confederation.