Trade is very simple, exchange of goods and services.
Commerce is bigger, it's part of business, it has transportation, insurance, logistics, etc.
Trade is a type of commercial activities.
For example, when you go shopping and bought a shirt, you have traded your currency for goods. We can't say that with commerce.
A commercial center (a shopping mall) is a center where many businesses practice trading and transport in/out their goods, they store them there or outside, they have to get a licence from a business authority (Chamber of Commerce), etc.