Commerce is a word that speaks of trade, or more precisely, an interchange of goods and commodities, usually between two countries. So, if one country gets oil from the Middle East, then this falls under commerce. Or if a country get soybeans. I think you get the picture.
Economics is the broader term that encompasses commerce, but other things that deal with finances. For example, the equities markets fall under the larger rubric of economics. So, does things that pertains to the internal state of a country, such as consumer spending.
Commerce is a term used to refer collectively to all the activities of business. It includes a range of activities such as buying, selling, manufacturing, transport, banking and insurance. Economics on the other hand is a field or subject of study. It studies how these various commercial activities are affected by the the attempt of a society or economy to make best use of its resources to produce valuable goods and services and distribute among different groups of people.
Of course we can study commerce also a an independent subject to learn how various types of commercial activities are carried out and to find ways of improving the efficiency and effectiveness of these activities.