Time's Arrow points in a definite direction. So we think. Or feel anyway.
I learned in my study of physics that the fundamental laws of nature do not indicate any such thing. Sean Carroll talks about this in his excellent lecture on the direction of time's arrow and the nature of the Universe.
Why do we remember yesterday but not tomorrow? In a word, tidiness. Specifically, the extreme tidiness that existed at the beginning of the Universe. Expressed in the language of cosmology, the extremely low entropy state of the Universe at the instant of the Big Bang meant that entropy (disorder) would always increase with time. Put another way, the fact that the Big Bang happened and that the Big-Bang-insta-Universe was a uniquely highly ordered and organized state means that the Universe would continually evolve into a more disordered and disorganized state over time.
One consequence of the continuous increase of the entropy of the Universe is that conscious beings (like us) perceive the future as different than the past.
Without the Big Bang, time would not point in one direction, and we would (if we existed in such a universe) remember tomorrow as well as we could remember yesterday.