The only meaningful difference between market-risk premium and equity-risk premium is scope. Both terms refer to the same concept and are calculated the same way. Yet the equity-risk premium only refers to stocks, while the market-risk premium refers to all financial instruments. Standard equity-risk premiums are usually higher than standard market-risk premiums on non-equity instruments. Generally speaking, equity is more risky than many other kinds of investments.
Due to their similarities, it is not uncommon for financial publications or pundits to use market-risk premium and equity-risk premium interchangeably. These concepts should not be confused with the historical market-risk premium – which is the same for all investors – and the expected or required market-risk premium for different investors.
Generally speaking, a risk premium is the incremental return of an investment or portfolio above and beyond the risk-free rate of return. Treasury bonds are normally held as the standard for risk-free returns.
To illustrate, suppose an individual has $10,000 to invest. She could place the money in Treasury bonds for a relatively low rate of return, say 2% per year. If she chooses the T-bonds, she has virtually zero chance of losing her principal.
The Treasury isn't the only institution that wants her money, though. Corporations offer bonds and stocks to raise capital, but they can't offer the same kind of safety that T-bonds have. They have to increase the offered return on their instruments to entice the investor to place her money with them.
The difference between the risk-free rate and the rate on non-Treasury products is the risk premium. When the non-Treasury instrument is a stock, the premium is referred to as an equity-risk premium. Any instrument can have a market-risk premium.