Why search new elements?
There are multiple reasons:


To improve our experimental prowess: There's no single catch-all process to create higher elements. While the basic process is to smash stuff together, once can't get much more specific than that. For each new element, the techniques used differ. And, as with any experiment, the complications which arise differ. Solving these lead to a deeper understanding of how to conduct experiments. Note that there have been gaps of years between the synthesis of various transuranium elements. This is not because scientists find it a useless task; it is because designing an experiment to synthesize these elements is hard.
To test stability: We have various nuclear models that can predict the stability of the elements. Creating new atoms helps validate these theories.
Islands of stability: Some higher elements are considered islands of stability in that they are more stable than their neighbors. If we can improve our experimental techniques to the extent that these can be generated, we may find some use for them. Of course, these may or may not be useful &mdash; these still are short-lived, just not extremely short-lived
Because we can!