Below are the examples I can think of about how a programmer and a non programmer brain differentiates;
First of all, a programmer brain should always think the worst cases and possibilities that other brains wouldn't think about. Ask a group of people about how to make a peanut butter and jelly sandwich and you'll point the programmer brain right away. It is about exceptions. You both have to see the whole picture and also think every little detail that can be go wrong.
Secondly, if you have a passion for programming, you could easily get obsessed with your code. For instance, if I had a bug that day or a design problem I couldn't find a final solution yet, the last thing I will think before sleeping will be that. I just can't take it off from my head; 'How can I make it better?', 'How could I prevent that happening again?' etc. (Of course my wife hates that because this is not a good topic for a pillow talk.) Many times I've experienced an enlightenment that the solution just formed in my head. I am not sure if I did sleep or awake. But I'd know what to do about the problem when I wake up in the morning. This is something comes with the obsession I believe.
Lastly, I remember when I was a child it was more important 'how stuff works' other than 'what it does'. That's why I've disassembled every electronic or mechanic toy I had during my childhood. When I find out computer games, I've thought about how do those things work after playing a few hours. I am just curious about it. Many millions of people don't even think how a cell phone works. Voice transferred over air? It is just like a miracle. But a programmer brain is curious, he/she thinks about internals. He/She even have empathy with the other engineers built those things. For example, I can easily find out how a setting is changed on any consumer electronics without checking the user guide because I can see the device from eyes of the engineers designed it; 'that setting should be there'.
There are many other differences, mostly originated because of programmer brain is more inclined to analytical thinking. And I believe most of them aren't related with talent but comes with experience and passion.