I discovered, while re-watching season one recently, that I too liked Tasha a lot more than I realized I did. I think she could have been used to offer a slightly unique perspective, coming from a "failed" colony to the Federation. I always saw her as being a little jaded, myself.
Had she stayed, we would have had no "Yesterday's Enterprise" the same way, or Sela. We'd have probably still had an altered "YE" and no Sela is a good thing by me. Worf wouldn't have been as overexposed, and might have gone to DS9 from the start, two more good things.
Someone (I don't recall who) once suggested that Yar might have developed a relationship with Riker, which I would have really liked to see. I don't think it would have necessarily lasted the whole series, but I think it would have been interesting, to be sure.