Why we can remember past but can't predict the future?

The ability to predict translates to entropy. The direction of prediction works along the direction of increasing entropy.

The reason we can remember the past but not the future begins with the arrow of time, defined by the increase of disorder or entropy. Stephen Hawking explains in Brief Answers to the Big Questions that our psychological arrow of time points in the same direction as the thermodynamic arrow because the functioning of our brain involves the increase of entropy [1]. 

Just as a tidy room naturally becomes messier over time, the universe moves from order toward disorder. When we form a memory, we create a record of the past, and this process generates a small amount of heat or mess in our bodies. Since the universe only allows disorder to increase in one direction, from past to future, this recording process can only work for things that have already happened. We cannot have a record of tomorrow because that would require creating order from the future's higher disorder, which goes against the fundamental laws of physics.

Even if the arrow of time did not exist, perfect prediction would still be impossible due to the Heisenberg uncertainty principle. To predict the future with absolute accuracy, one would need complete knowledge of the position and motion of every particle in the universe [2]. 

To put it into context, consider that you are playing football. The future of the football, meaning where it will move and how it will behave, is determined by its current position and momentum. If you could capture an absolutely perfect snapshot of both values at any single moment, you could theoretically calculate exactly where the ball will be at any future time using the known laws of physics. However, the uncertainty principle states that we can never know both the exact position and exact speed of a particle at the same time. This is not a technological limitation but a fundamental property of quantum mechanics. If the laws of physics are like a computer program, the uncertainty principle means we are forever forced to feed it low quality or incomplete data about position and momentum, which will always give us unsatisfactory results.

References

[1] S. Hawking, Brief Answers to the Big Questions, 1st ed. New York, NY, USA: Bantam Books, 2018.

[2] J. W. Grove, "It's about time," Queen's Quarterly, vol. 109, no. 1, p. 120, 2002.