Under what conditions should the External Rate of Return (ERR) be used instead of Net Present Worth (NPW) and vice versa? I have just read the concept of both Net Present Worth (NPW) and the External Rate of Return (ERR). This is what I have understood of these two values:

NPW is the net discounted value of all the cash flows that occur throughout the life of project, while the discount rate is the MARR -- a rate that I have decided as a minimum rate of return for acceptance of a project. If NPW > 0, accept the project else reject.

ERR is the rate of return of the project when I reinvest the cash produced at a rate RR (Reinvestment Rate). This rate may be different than the ERR itself. If ERR > MARR, accept the project, else reject it.

I may have understood wrong, please correct me if I am wrong. 

So both of the methods are used to make accept/reject decisions for the projects. Certainly there are perks of using one over the other, otherwise there would not be two distinct methods. Then under what conditions is one preferred over other? Or is one of the methods always better than the other in general? Why?
Investment protects are inherently risky, and when calculating either of these numbers many assumptions need to be made. From projecting what future inflows and outflows will be, to deciding what rate of return to use to discount these projects payoffs. As far as I know, usually both measures are calculated and considered when making a decision since they give a slightly different information. 

For example, the NPW needs to assume some rate at which to discount payoffs, in your example you call it the MARR, the choice of such rate can be controversial, while the calculation of the ERR does not need to assume any rate. For example I can show you a project with a very high NPW, but this high number may only be due to the fact that I chose a very low MARR, this will be evident because the ERR will necessarily be low. The two measures give a more complete picture. Likewise, I can give you a project that has a very high ERR, but the actual level of the value of the project can be tiny, this will be reflected by the fact that the NPW will be a small positive number.

If the MARR is not controversial, the king of all measures is certainly the NPW, as it literally tells you how valuable is a project. But there are many ways to tinker with the NPW and make it look better so complementary measures give you a better picture of the quality of the project.