In a parametric model, you know which model exactly you will fit to the data, e.g., linear regression line. In a non-parametric model, however, the data tells you what the 'regression' should look like.

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Let me give some concrete examples.

Parametric Model: yi=β0+β1xi+eiyi=β0+β1xi+ei

Here you know what the regression will look like: a linear line.

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Non-Parametric Model: yi=f(xi)+eiyi=f(xi)+ei

where f(.) can be any function. The data will decide what the function f looks like. Data will not tell you the analytic expression for f(.), but it will give you its graph given your data set.

The reason why people say that there is inherently no difference between parametric and non-parametric regression is that the function f(.) can be perfectly approximated by an infinite-parameter model, which is parametric.