What is the purpose of designating the bottom number in some cases?

For example. Couldn’t something played in 4/8 time sound exactly the same as something played in 4/4 time? I’m talking about if the tempo is matched and everything.

I mean, if there are 4 beats to the measure, then what difference does it make what the note value is? Each measure you hear 4 beats so it’s all the same isn’t it?