To add to what Dennis said - With the .015 DIA solder you do have better control on the amount of solder that is applied. You will have to feed a longer piece of solder into the joint do to the smaller diameter solder having less volume per inch. It feels you are using more solder but you are not. (Which weighs more? A pound of 0.031 DIA solder or a pound 0.015 DIA solder?)
It's not so much a matter of weight as a matter of time. The longer you are feeding that thin solder into the joint, the longer you are keeping heat on the joint and th components. If you are using flux core solder then you are probably not getting the proper amount of flux on the joint either if you are using thin solder on a large joint. They make different size solders for a reason and it really is best to use the correct size for the joint that you are soldering. I have about 3 different sizes with .031 on the stand and .015 near by. I don't remember the size of the third one but it is a heavy solder that I use for large components and heavy wire.
I don't not understand your difference using the 031 or 015 solder sizes.
Let me explain, the issue is enough heat and solder to make a proper solder joint. Using a large diameter solder means you have to keep the soldering tip longer on the joint because it takes some time for the heat from the tip to melt the solder and heat the joint for the solder to flow properly.
Using smaller solder diameter your have the tip still there and yes the solder will melt quicker as the volume sizes of the solder is smaller, but you still need the proper amount of heat on the entire solder joint to make it flow properly, adding a little more small diameter solder satisfies the heat needed on the solder joint.
Either way using the larger or smaller diameter solder on the regular size solder pad will work, it is having enough heat for the flux to clean the area, the solder to flow properly and the right amount of solder to make a good solder joint.
Solder tip size and temperature must be correct to make either size diameter to work correctly. So the recommendations on this site for soldering irons and temperature are good.
It takes some practice to get into the rhythm on making good solder joints and you will know when you are applying enough heat and solder to a joint.
I will get off my soap box now.