The latter one allows you to write formatted output.

Response.write - it writes the text stream 
Response.output.write - it writes the HTTP Output Stream. 

They are used to display output text in formatted manner as 
below
Response.Write() output fotmat :
Response.Write"<h1>" & iTimer & "</h1>"
Response.Output.Write() output format :
Response.Output.Write("<h2>Process running as {0}</h2>",
    WindowsIdentity.GetCurrent().Name);

R.W:: It writes to  the browser
R.O.W :: It writes to the file

response.output.write allows String.Format style output and 
the response.write doesn't

For example : Response.Output.Write("{0:d}", "Current Date 
Time is: ",DateTime.Now);