The 1907 dime according to my scale is 1g and the 1903 is 2g. 1907 is also thinner. Same width as the 1903.
My guess would be the extreme wear that would cause the discrepancy in the weight. I have a very worn Barber dime that I dug up while metal detecting that is barely half the thickness of a "normal" one.
Per your pictures, both dimes appear genuine. I strongly suggest you get a better scale if you're going to use it for this purpose. What is the resolution of your scale? A good scale for coin collecting, and for helping to determine authenticity, will measure down to 0.01 gram. I'm assuming your scale is not quite this accurate.
Yah. The coins just wore this way. They have roughly the same value, regardless of the wear pattern. If you want to see the difference a good scale can make, take them both to a friendly neighborhood jeweler, and ask them to weigh them both. You'll likely see that they're not quite so far apart as you thought.
Based on your reported weights - both coins would be fakes. Correct weight is 2.5 grams. If weighed on an accurate scale what you will likely find is that the '03 dime will weigh between 2.45 & 2.5gm and the '07 dime between 2.35 & 2.5 gm.
I agree with the others, I suspect your scale is not accurate. Try weighing some recent Zincoln cents. They should weigh about the same as a silver dime so see what your scale says about them.
