There's nothing mysterious it's just one key being used for two different but close notes. It's the same as if you were told to paint one wall sky blue and another wall baby blue, and you used light sky blue to paint both hoping no one will notice.

The no-bullshit explanation of this is:

C#4 is defined as 2^-8/12*440 Hz or about 277.18 Hz.

In the key A, the music note A3 has by definition a frequency of 220 Hz. A major third above that has by definition a frequency of 220*5/4 = 275 Hz. We notate this as C#4 even though that corresponding piano key string intended to be played when we write C#4 on the music sheet, is tuned to ~ 277.18 Hz.

In the key of Ab, take the note Ab3 which has by definition a frequency of 2^-13/12*440 Hz = ~ 207.65 Hz. Then, a perfect fourth above that, called Db4, is by definition about 207.65 * 4/3 = 276.87 Hz. When we write Db4 we mean to play the piano key tuned to 277.18 Hz, although this is not the frequency we intend.

So in both cases you can see that the frequency we want to play and the frequency that we can produce from the piano key are different. We want the notes to be in an exact ratio of 5/4 in the first case, and similarly in the second case we want some exact ratio, but we cannot get it with a standard piano it's impossible. There's nothing special about C# and Db, there are a lot of notes that we do this sort of rounding thing, otherwise we'd have like 70 different names for notes and a lot would be like 0.1 Hz from each other so there'd be no point. The piano key sounds the same no matter what intention you have in hitting it, although some people may be able to placebo themselves into hearing 2 different notes. So what he means by "sounds different" is that we use the same key when really we should take out the C# key and replace it by 2 keys: one for when we play a major third above A and another key for when we play a perfect fourth above Ab. That is all there is to it.