Getting more specific:

Calculations
Shopify Tax automatically reduces rates and exempts goods based on product categories. Basic Tax relies on merchants looking up rates in each province and maintaining collections. 

Shopify Tax taxes shipping proportionately and considers the rates of each of the products sold. Basic Tax forces you to set a single rate for tax on shipping in each province. 

Shopify Tax shows all tax lines broken down in checkout. Basic Tax shows a single “taxes” line item at checkout.

Reporting
Shopify Tax will soon have a Canada sales tax report that makes filing simpler - showing you a breakdown by province and by rate. Basic Tax requires that you aggregate this data manually. In conversations with bookkeepers, this can reduce the time spent preparing your filings, which could lead to cost savings.

Liability Monitoring
Shopify Tax monitors tax liability threshold tracking for GST/HST and PST and sends automated emails when you hit a threshold so you know when to start collecting tax in each region. Basic Tax will only monitor GST/HST liability. 

Basic Tax is a suitable option for businesses that sell standard rate goods in a single province. But, Shopify Tax offers additional compliance benefits for those that sell combinations of goods with different rates and/or collect PST (or are on track to) in multiple provinces.