Why develop Edward curve formulas that deviate from unification? Edward curves were considered initially because they provide a unified formula for both doubling and addition, thus having inherent side-channel resistance. But a lot of work has been done recently involving construction of a different class of addition and doubling formula (e.g. differential addition formula, w-coordinate formula), particularly concerning efficiency of these formulas in terms of the number of field multiplications/squarings. Thus, the point of unification becomes invalid.

Please can anybody explain/cite some relevant papers which address this issue?
Edwards curves can be implemented using a unified formula for addition and doubling; i.e., one can implement addition such that
$$\mathrm{dbl}(P)=\mathrm{add}(P,P).$$
Performance wise it is however more efficient to consider both functions separately, since the doubling can be implemented more efficiently than the addition. 

Depending on the representation and the formulas, the unification property will be preserved or not. For instance, in "Twisted Edwards curves" by Berstein et al., there is a formula for $\mathrm{add}(P,Q)$ and a faster one for $\mathrm{dbl}(P)$, but the above property still holds. In "Twisted Edwards curves revisited" by Hisil et al., there are also two separate formulas, but the above property doesn't hold anymore.

I think the unification property looks great in a paper as it makes the description cleaner; however one does not necessarily need it in practice. Some examples:


Example 1: "Scalar multiplication (based on Exponentation by squaring)". When the factor is lower than the order of the group, you know that the next doubling is always higher than the accumulated value, so that an addition will always take place between two different points. 
Example 2: "Addition of random points". The order of the group is in general so large, that the probabilty that two randomly chosen points are equal is negligible. 


So in many situations the absence of the unification property does not seem to be an issue. It will of course depend on what precisely you are doing with the curve.