As I faced before with these two (trade and business) words I understood there is not any different between them, but when I searched in English to Persian (my language) dictionary for exact meaning of them, I found some difference. In addition I should say I haven't trust to this dictionary because it showed me some mistake meanings before.
Both words have multiple different, often overlapping, but rarely identical meanings, so a short answer is difficult.
Business in this context is a noun usually referring to a legal entity that carries out some form of activity, usually to make money (this can be called business activity). Software consultancy and plumbing services are both examples of businesses.
A trade is something that describes the skillset you use to earn a living. Software developer, Plumber, Painter, etc. can all be described as trades.
In some countries (Australia, probably many more) trade also refers to a specific set of qualifications. In our case, it means a tertiary qualification in a typically technical field commonly obtained through a TAFE, which for which the student did not have to attend university. E.g. A plumber is a tradesman (ie. they have a trade), but a Software Engineer is probably not (as they likely would have attended a university). However, colloquially it would still be correct to answer "What's your trade?" with "Software Engineer" because in that context we are probably referring to its meaning as "skillset".
Completely apart from these descriptions, both the words business and trade have other overlapping meanings. For example, to trade is a verb which describes the exchange of goods or services. To do business means to perform some sort of business activity (usually with at least 1 party aiming to turn a profit), which might involve trading goods. The act of trading, and (now the noun) trade itself are usually part of some sort business activity.
Here's a paragraph that utilises the uses of the word I just described:
You cannot carry out a business activity without conducting some form of trade. Whether you run a plumbing service or are the CEO of a large corporation; your business must exchange its goods and services for either money or other goods in order to remain profitable. These exchanges are trades, although we don't usually refer to the exchange of goods for money to be a trade (we colloquially call it a sale, even though it is technically also a form of trade). In doing so the business might (but doesn't have to) employ the skills of tradesmen, because these individuals' trades are valuable skillsets that customers are willing to pay for (e.g. you might pay a plumber to fix a leak in your home). Whenever such a corporation exchanges its services for the money of customers, it can be said to be doing business.
"Trade" in English has two meanings: 1) as a noun: a profession (as described by Quant); 2) as a verb (to trade) or noun: the process of buying and selling; the exchange of goods (often physical goods, but it doesn't have to be). Iran's export of oil in exchange for money is an example of trade.
"Business" is a more general term. It is always a noun. It can mean: 1) the legal entity that engages in economic activity ("He works for a business.") 2) the process of conducting economic activity ("He is conducting business.")
Trade (in the sense of buying or selling) is a form of business, or more precisely, something businesses do, but it is specifically refers to the exchange of money (or other benefits) for goods.
Most languages make a distinction between the two terms. I suspect that Persian does too.