What's the difference in the interpretation of $y = x$ and $y(x) = x$? Both $y = x$ and $y(x) = x$ describe a line, but what are some of the differences between their uses, and can they ever be used interchangeably?
$y=x$ simply represents a line passing through origin, while $y(x)=x$ here $y$ is behaving as a function of $x$, in this case your function is identity function , you have to specify a domain and a codomain. Other possible functions are like $y(x)=x^2$ and so on for example.
The only reason y(x) = x describes a line is because, for any value of x, y(x) will be the same, as per the definition of y(x). As Kushal Bhuyan said, you can define y(x) to be any function you want. Also, when you graph y = x, there will be an x - axis and a y - axis, while if you graph y(x) = x, there will be an x - axis and a y(x) axis (or any other function that you choose).