From a 10 second Google search...
"The issues are that the two cartridges work at different pressure levels, with the 357 Sig operating at higher pressure than the .40. The 357 Sig case has a thicker web, and is thicker at the junction of the wall and web of the case. The 357 Sig is also longer.
There are people who neck down .40 brass and use it, but it's not recommended for the reasons I've listed above. The 357 Sig case was engineered from the ground up for that caliber.
If you choose to do this, then you would have to keep your pressures low, and there will be a problem with the even shorter neck of the necked down .40 case. The 357 Sig neck is short enough as it is to cause some people problems with keeping the bullets from being set back during cycling of the action, and using even shorter necks would exacerbate that problem.
These days there is plenty of proper 357 Sig brass available, so my suggestion would be to just buy the proper brass, if you decide to do the conversion. I have two pistols in 357 Sig, and I've loaded about 15,000 rounds of it. This is one of those rounds that you don't want to cut corners on."
Hope this helps.
Personally I have seen .357 Sig brass sell for about $30-40 per 1000 of used range brass. At that rate I would just buy it, inspect it and develop a load that works in your gun. Plus converting brass sucks, I did about 500 for my 9mm mak a few years back and its painful to loose the cases after all that work, I can't imagine how the 300 BLK people feel.