Piecewise regression vs multiple models Say you have a piecewise regression model with one breakpoint. Is there any difference between that and just creating a separate simple regression model for each segment of the IV? 
 A: To add to the other answer, there are (at least) a couple other major differences. The piecewise regression segments will connect at the knot (see this link). When fitting two separate regressions, the regression lines may not touch at the point where one segment ends and the other begins.
Another difference is that with the piecewise regression, each model parameter is being estimated using the entire dataset, and thus you will have just one output table with coefficients, p-values. MSE, etc. With two separate regressions, each segment’s regression only uses data from that segment, so you’ll have two output tables to interpret, which leads to other consequences surrounding degrees of freedom and causes issues when trying to estimate overall model statistics like $R^2$, lack of fit F-test, etc. 
A: The mathematics for fitting a piecewise linear regression is very different to fitting two independent linear regression methods.
Let's ignore mathematics. Practically, the most obvious difference is that piecewise linear regression estimates the breakpoint for you while fitting two linear regression require your visual inspection. It will be tricky for visualization if you have more than a single independent variable.
