Shares vs. Stocks: An Overview
The distinction between stocks and shares in the financial markets is blurry. Generally, in American English, both words are used interchangeably to refer to financial equities, specifically, securities that denote ownership in a public company. (In the good old days of paper transactions, these were called stock certificates). Nowadays, the difference between the two words has more to do with syntax and is derived from the context in which they are used.

Similar Terminology
Of the two, "stocks" is the more general, generic term. It is often used to describe a slice of ownership of one or more companies. In contrast, in common parlance, "shares" has a more specific meaning: It often refers to the ownership of a particular company.

So if someone says they "owns shares," some people's inclination would be to respond, "shares in what company?" Similarly, an investor might tell their broker to buy 100 shares of XYZ Inc. If they said "buy 100 stocks," they'd be referring to a whole panoply of companies—100 different ones, in fact.

That comment "I own shares" might also spark a listener to respond even more generally, "Shares of what? What sort of investment?" It's worth noting that one can own shares of several kinds of financial instruments: mutual funds, exchange-traded funds, limited partnerships, real estate investment trusts, etc. Stocks, on the other hand, exclusively refer to corporate equities, securities traded on a stock exchange.