Basically a smart grid differs from what we now have because it is capable of distributing electricity more efficiently and of allowing communication between various parts of the grid.
What we have now is a system where the electricity gets dumped into the grid and sent out around the country. It's essentially a very crude process with no information going back up the grid to utilities. This means, for example, that many power companies don't even know when something has gone wrong until customers call to tell them the power is out. With a smart grid, there would be sensors that would tell the companies what is going on and reroute power flows in efficient ways.
Smart grids would also allow give consumers more information. We would be able to know, in real time, how much electricity we are using, what we are using it for, and how much it is costing us. We would be able to know when electricity use is high or low and adjust our usage accordingly. The power companies could then charge different prices for electricity at high and low usage times and we could save money by using it at low times.
Basically, a smart grid would be more like the internet, with all sorts of communication and information going in all directions than the current system, which is more like water being dumped in pipes and sent out.