There are several things you need to be aware of when doing this.
First, you'll incur the cost of getting new rims (which I'm sure you already realize). This is actually pretty common when fitting snow tires, though. Having a second set of cheaper rims (steel is cheaper than alloy) you can change out yourself versus having to take them to the shop to get done is a large cost savings, not only in money, but time as well (do them at home at your convenience).
NOTE: When you take one set of tires off, ensure you mark them for placement on the vehicle. Tires should always rotate the same direction or wear will increase tremendously. If you just write on the tread area with some chalk the position of the tire when it comes off the car, you can put it right back where you got it without issue. (ie: DF=Driver's Front; DR=Right Rear; PF=Passenger's Front; PR=Passenger's Rear)
Secondly, you need to see whether your car can actually fit smaller rims. It is quite common for there to be interference between the brake rotor and the smaller rim. A side consideration when getting your new rims is that you need to ensure you have the correct offset. Not just any rims will do, they must stick out the correct (enough) distance or the new rim will interfere with body and brakes. There is a lot to put into consideration here. You will also need to make sure the size of the tire will fit on the rim (tire width and what it needs to go onto).
Thirdly, when considering a snow tire, make sure the rolling distance or "rotations per mile" (RPM) is the same or nearly the same between the two tires. Having a huge difference between the two tires will cause your speedometer calibration to be off. It will also affect how the vehicle behaves (ie: a taller tire will require more torque to get a vehicle rolling and stopped; a shorter tire will cause worse gas mileage on the highway because the engine has to rev higher to achieve the same ground speed.) Most major online tire stores (like tirerack.com) have this or at least the diameter of the tire. You can compare the two and make a decision from that.
With all that said, you shouldn't have an issue with running the smaller tire, especially considering the small amount of difference there is between the two.