Weakly relatively hyperbolic groups A finitely-generated group $G$ is weakly hyperbolic relatively to a collection of subgroups $\{ H_1, \ldots, H_r\}$ if the graph obtained from a Cayley graph of $G$ by coning off the cosets of the $H_i$'s (ie., adding a vertex for each coset $gH_i$ and linking this vertex by an edge to all the vertices of $gH_i$) is hyperbolic (in the sense of Gromov). 
This definition is due to Farb and many groups turn out to be weakly hyperbolic relatively to some collection of proper subgroups. However, it seems that too many groups are weakly relatively hyperbolic, so that no algebraic properties may be deduced; see for example this article. On the other hand, being relatively hyperbolic, as defined by Gromov, is a strong property. 
In fact, I realise that I have never seen any property of a group proved by using its weak relative hyperbolicity. Do you know such an example, or loosely speaking:

May the weak relative hyperbolicity be useful?

 A: Weak relative hyperbolicity all by itself may not be so useful, but it proves useful in combination with other things. Strong relative hyperbolicity is an example, but there are weaker examples. 
For instance, the action of a surface mapping class group on the curve complex of that surface proves weak (but not strong) hyperbolicity of mapping class groups, which is a result of Masur and Minsky. The fine study of that action, coupled with associated actions of subsurface mapping class groups on their curve complexes, yields a very powerful theory, the "Masur-Minsky hierarchy theory".
Other examples include the whole of Bass-Serre theory, which can be summarized to say that any group which has a nontrivial graph of groups presentation acts on its Bass-Serre tree. With a few more hypotheses on the graph of groups, one sees that these groups are weakly hyperbolic relative to their vertex groups (strong relative hyperbolicity holds under somewhat stronger hypotheses, for instance if all edge groups are finite). And these actions are incredibly useful.
