This discrepancy is almost entirely a function of living space size and the cost to heat or cool this space.
Where I live, 100 sq meters would be a fairly average or small studio (meaning one room with integrated kitchen and separate bathroom/closet) or one-bedroom apartment. I doubt apartment energy usage per sq/meter in the US vs Germany is very different at all.
The real difference comes down to houses. My house is right around 270 sq meters, with a disconnected basement (outside access, though it is heated and cooled) and two living levels, two potable natural gas water heaters (also used for heating) and two HVAC systems. My house was constructed around 1995 and still has all original equipment. The windows are poorly insulated. I run many LED lights, no CFLs, and some incandescents (bathrooms). I have a 55" tv, two desktop computer, one NAS, two laptops and an array of smaller rechargeable electronics. Clothes washer and dryer (both electric), refrigerator, and an extra freezer in the garage are included. Out of curiosity, I just pulled my energy usage for the last year (2014-02-01 through 2015-01-31). Total electricity usage from the electric company was 10,067 kWh. The biggest single month was July when I used almost 1600 kWh on air-conditioning.
So, on the face of it, I use almost 3x the energy you do in a year! I'm one of those pig Americans!
As a rough calculation, your flat gets 3500 kWh / 100 m^2 = 35 kWh / m^2.
My house gets 10,067 kWh / 270 m^2 = 37.2 kWh/m^2
What's that--a 3% difference when you take into account the area?
In all honesty, when I ran these numbers, I was very surprised to see this. I was expecting that you would be far more efficient per area, but the difference is really inconsequential. Is there some error in your numbers (or my math) that I am missing?? The much larger number on my side really just comes down to my much larger space. In fact, if you include the fact that I have 5 people in my space, I think I would be considered more efficient! Americans still do tend to have larger families than Europeans (that's true when comparing immigrant populations in both regions and when comparing non-immigrant populations).
I also pulled the natural gas numbers for my house for the year 2014 (I couldn't get the exact same range) and total usage was 735 CCF (centum cubic feet). I'm not sure what you measure natural gas usage in--m^3? If so, approx 20.8 m^3. I don't know this would compare to you.
Americans a generation ago were used to much smaller houses, and Europeans today are certainly used to smaller spaces and, I should note, smaller families. The reason we moved to our current house (our previous house was ~140 sq meters) was with the birth of our 3rd child, we wanted more space. Not everybody wants to live in the same size space. I'm happy to pay more for the difference.