Well, one reason is if I want to accurately infer past events from present memories. E.g., if I want to know where my keys are, it is helpful to know how where my remembering self (hereafter RS) remembers putting my keys relates to where my past experiencing self (ES) actually put my keys.

Another reason might be if my "fleeting" ES's experiences actually have longer-term consequences outside of my RS's recollections. E.g., if I am experiencing anxiety in the presence of clowns as a consequence of some experiences my earlier ES's had which my current RS doesn't remember, I might still prefer to not experience that anxiety; similarly, if my current ES's experience may create anxiety in some future ES I might care about changing that experience for the sake of my future ES. (As well as, of course, for the sake all the future RS's who remember the anxiety uncomfortably, despite not remembering the event that causes it.)

But let's back this up a little. Never mind whether you should care for a moment... do you care about what you're experiencing right now? Or do you care more about what you will remember having experienced right now?