Writing the query on paper requires the candidate to demonstrate the conceptual knowledge of the SELECT-FROM-JOIN-WHERE-ORDER BY construct without fumbling through it by trial-and-error at the keyboard (smart people with weak coding skills can usually manage to cobble something like this together if they can code-execute-evaluate enough times).
May be, if you are hiring graduate...
Some prefer "smart people" over "strong coding skills", simply because if you smart, you can learn...
Also, from the business prospective, if I ask someone, "how to solve Fizz Buzz puzzle in T-SQL", I would expect first choice from real expert as:
Google for "Fizz Buzz T-SQL".
It will, most likely, in real life, help to solve the common problem a lot quicker, meaning saving me time and money!
In my humble opinion, that small thing constitutes difference between Professional and Expert. Yep, good professional can write the right code in 5 min. However, expert can spot common problems and find ready-to-use solution in a matter of seconds (it's all to do with inventing bicycles)...
Then I could discuss other, more SQL relevant things and even insist he will scramble something on paper, still putting candidate at front of SSMS and ask him to use it would give me better idea of his skills and experience with SQL Server...
Even simple "can you tell me what columns "this" table have and what their datatypes are?" can give you good insight of how experienced the candidate is. Some will use mice and expand the table in Object View (would be good to ask for some table which is located at the middle of the 1000 odd list), some will use sp_help, some will press couple of buttons and get what you need...