Let's say you have 2 algorithms. Constant time O(1) and exp time 2^n. It seems constant time is better. But in fact it runs always 42 years. While exp one starts to run longer than millisecond for sizes bigger than number of particles in universe.

Big O says how fast complexity grows, but it doesn't say what that complexity exactly is.

Does something like that happens in real world? Yes, but of course not in such drastic ways.