Investors who are themselves Canadian and already hold Canadian dollars (CAD) would be more likely to purchase the TSX-listed shares that are quoted in CAD, thus avoiding the currency exchange fees that would be required to buy USD-quoted shares listed on the NYSE.
Assuming Shopify is only offering a single class of shares to the public in the IPO (and Shopify's form F-1 only mentions Class A subordinate voting shares as being offered) then the shares that will trade on the TSX and NYSE will be the same class, i.e. identical. Consequently, the primary difference will be the currency in which they are quoted and trade. This adds another dimension to possible arbitrage, where not only the bare price could deviate between exchanges, but also due to currency fluctuation.
An additional implication for a company to maintain such a dual listing is that they'll need to adhere to the requirements of both the TSX and NYSE. While this may have a hard cost in terms of additional filing requirements etc., in theory they will benefit from the additional liquidity provided by having the multiple listings. Canadians, in particular, are more likely to invest in a Canadian company when it has a TSX listing quoted in CAD.
Also, for a company listed on both the TSX and NYSE, I would expect the TSX listing would be more likely to yield inclusion in a significant market index—say, one based on market capitalization, and thus benefit the company by having its shares purchased by index ETFs and index mutual funds that track the index.
I'll also remark that this dual U.S./Canadian exchange listing is not uncommon when it comes to Canadian companies that have significant business outside of Canada.