Finite representations and programming languages Countably inifite
Here is a simpler situation highlighting the difference. The set of finite binary strings is countable. The set of infinite binary strings is uncountable.

Another example: the set of numbers with finite decimal expansion is countable. The set of numbers with infinite decimal expansion is uncountable.

The reason that the number of languages is uncountable is that you have infinitely many choices: for each word, you can decide whether it's in the language or not. That's why it's like that infinite binary string considered above.