Asymptotic notations are generally used for large values and we mainly use big-O notations.
Because when we do it for large values then after a certain point $c$ graphs becomes constant and its easy to choose which algo to use.
For eg:- see this question https://gateoverflow.in/2466/gate1994-1-23
After the point c=10000 g1 will perform better than g2, since g1(n)=O(g2(n)) after 10000.
We dont do it for small inputs because the graphs may vary.
like in example above g2 was performing better till n=100 but after that g1 and g2 both were performing same till 10000.
also for smaller inputs the time-space complexity do not matter much. I mean like when we are talking like
X takes 120 hours and Y takes 110 hours for large inputs
and saying that X takes 5 seconds and Y takes 50 seconds for smaller inputs does not matter much.
we will choose Y. (since Y is more effecient for bigger inputs).