The simplest explanation is that the two systems are not compatible so Canada protects their side via a large tariff wall.
Canada uses Supply Management boards to control the price of dairy (among other things). These boards issue quotas to producers. For instance, a dairy farmer may own a quota of, say 50 kg of milk fat. Thus he could produce milk until the total amount of fat contained in that milk reaches 50kg. At that point, any extra milk cannot be sold. The quota can be bought and sold. I think the current price is something like $24K per kg in Ontario.
The total amount of quota is closely matched to the actual demand of the Canadian domestic market. This ensures a stable price for dairy since there is neither over nor under production. The dairy farmers are assured of a known price for their product and consumers are assured of a stable supply.
In contrast, the USA uses a more open system with no controls. This tends to lead to boom and bust cycles since production is not matched to demand. Instead, the profit motive drives farmers to produce as much as possible. This can lead to lower prices for consumers, but an unstable supply. During bust periods, many dairy farmers are driven to bankruptcy or have to sell off their herds. To help with this, the US, like many countries, uses subsidies, both direct and indirect, to help keep the farmer solvent. While admirable in intent, it also tends to extend the bust cycle since it keeps more cows in production and lowers the milk price.
Because the Canadian system produces milk at a fixed, relatively high price compared to subsidized US milk, the US product would quickly flood the Canadian market. So large tariff walls are used to keep US dairy out (more later).
This is the basic explanation of why Canada charges 270% duty on dairy. Now, a bit more detail.
When NAFTA was negotiated, the US was angry about the trade walls. As a concession, the Canadian government agreed to allow up to 10% of domestic demand into the country duty free. Only after that would the 270% tariff rate apply. In contrast, the US allows up to 3% of dairy into the US and puts up high tariffs after that. The rates are not as high as the Canadian rates, but they don't have to be because the Canadian product, being unsubsidized, starts at a higher cost. Because of this, the US runs a large surplus in the dairy trade with Canada.
As stated, Canadian dairy is unsubsidized. Estimates of US subsidies vary but the most common I've seen is around $22B per year which is about 73% of the market. So in Canada, the consumer pays for the system directly through higher prices while in the USA, the consumer pays lower direct prices, but must also pay the bulk of the cost indirectly through taxes.
Aside from tariff walls, there are other barriers. For instance, Canada does not use growth hormones on its dairy cattle. With stable pricing, there is no need to. The US farmers, anxious to increase production, do use hormones (not necessarily everywhere). Thus Canada has banned milk products containing hormones.
The USA is not alone in their anger with the Canadian supply management boards. Europe also heavily subsidizes their farmers and wants access to the Canadian market. In a concession to get access to the EU, Canada has recently agreed to allow cheese in duty free. Thus Canada, while anxious to protect their industry, is willing to negotiate if it can get something in return.
Finally, regarding tariff rates in general, the USA has higher rates than Canada. The 2016 weighted average for Canada was 0.8% while the US was 1.6%. This is because, although we have high tariffs for dairy, the US has high tariffs for things like tobacco or peanuts. Some tobacco products have tariffs as high as 350%.
So in order to drop Canadian Supply management and the associated tariffs, the industry would have to be subsidized to the levels of the US and Europe. Once that happens, the US would still have trouble selling in Canada. I rather suspect they would sell less since it would also render the 10% duty free exemption valueless. In addition, our milk prices would be so low that we in turn could invade the US market! So what Trump is demanding would hurt the Canadian dairy industry AND the US industry.