I can see that you can do this using SQL server but I fail to see any benefit to using SQL Server rather than wirting some code using a normal computer language like C.
Surely it is much slower using SQL Server?
First, I take offense at your characterization of SQL as not a "normal computer language" -- unless, of course, you mean that it's above normal. Second, I laugh at your characterization of C as normal in any way. 😛
On a serious note, sometimes programming isn't about performance, or even about successfully accomplishing a task, but about learning. From the introduction to the article, it's clear that Silvia wasn't trying to create the most efficient or useful neural network, but more interested in how two programming concepts (a relational database and a neural network) might be brought together.
Also, bear in mind that many organizations can justify a SQL programmer where they can't justify a .NET or C programmer -- and they may benefit greatly from a neural network in SQL.