1. You know that restaurant round/around the corner? It's closed down.
2. You know that restaurant round/around the corner? It's closed now.
(Are both sentences grammatical? Are both the same? Do we have to use 'round' or'around'? Are both OK?)
Both are the same, yes.
The word "around" the correct choice. Some people use "round" but it's more of a vernacular choice, I think.
http://www.answers.com/round
http://www.answers.com/around
This pair reminds me of "till" and "until" -- two very different words that are often used interchangeably in less formal speech.