Significant difference between 28C and 32C tires? I have a Schwinn Sporterra 2011. It has a 622x20 rim and the default tires are 700x38C. I'm thinking of "upgrading" to thinner tires. 

Now, my local cycle shop has Specialized tires that are 32C. They can also get me Schwalbe 28C tires, but in 2-3 weeks time. What I'm trying to understand is, coming from 38C tires, how much of a difference will 28C or 32C make -- Difference to things like speed and handling? Will 28C be much "faster" and worth the long wait time?

(I'm actually thinking of switching to a road bike in 6 months but till the time I save up for that, I want to get thinner tires on the current bike. I was told they'll make quite a difference. Thoughts?)
They will make a significant difference, but the size of the tire alone won't make much of a difference to your speed. If your current tires have a substantial amount of tread on them, you may notice a speed difference since smaller tires tend to have less tread on average.  It's probably not going to be a substantial difference though, even if you have the knobbiest 38's on the market. Your biggest speed limitation is wind resistance, and while the narrower profile of the tires will knock that down a little, it's a small portion of the overall wind resistance that you experience. 

Where you'll notice the biggest difference will be in how much harsher the ride will feel with skinnier tires. They run at higher pressures so you end up feeling every bump in the road more.

You'll also notice that the steering feels more nimble due to the lighter weight of the skinnier tires.

You'll also be more prone to pinch flats with skinnier tires. To what extent will depend on the tire pressure that you've been running and the tire pressure you're going to run.