Converting VARBINARY HEX to VARCHAR
You need to have a distinction between varchar and nvarchar. There is a big difference. If I try to create varbinary from your "should work out to" value:

SELECT CONVERT(VARBINARY(64), '201407240005688309')


I get this value, which looks kind of like the value you're converting, but not quite:

0x323031343037323430303035363838333039


So now if I change the input to be nvarchar:

SELECT CONVERT(VARBINARY(64), N'201407240005688309')


Now I do get the value you wanted:

0x3200300031003400300037003200340030003000300035003600380038003300


So if I take that value and try to convert it to nvarchar instead of varchar, we're back where we started:

SELECT CONVERT(NVARCHAR(MAX),
  0x3200300031003400300037003200340030003000300035003600380038003300300039)


Result:

201407240005688309


The online converter you're using must be using Unicode behind the scenes. You need to do that, too.