This is called light rail because of its limited capacity compared to a traditional subway, or heavy rail system. Seattle's light rail system runs 2-car trains, which is already a comparable size boost compared to a bus. However, a subway or heavy rail system will run 6- or 8-car trains, such as the systems in New York or Washington, D.C. Most subway systems also tend to run at higher frequencies.
In short, the "light" and "heavy" mostly to how many people the rail system can carry, or capacity. Larger trains at higher frequencies will carry many more passengers than smaller trains at lesser frequencies. There are also more, less important details. If Seattle's light rail system was all underground, you could call it subway, but subway has started to refer more to heavy rail than underground.
So why isn't Seattle ready for a full-fledged subway, heavy-rail system? Apart from the unwillingness of people to pay for such gargantuan projects and the geological problems associated with building infrastructure in some areas, there simply aren't enough people in Seattle at adequate densities to make such a system efficient and worthwhile.
Will this change, however? Yes, absolutely, and quite rapidly. The reason the station platforms for the Link are so big compared to the trains is in expectance of a large increase in density and people along the line that was created. This is a good thing. Seattle is increasing rapidly, and areas like South Lake Union, Capitol Hill and the University District already have subway-ready populations.
Many areas of Seattle are also growing so much that they will be ready for heavy rail sooner rather than later, like Uptown, South Lake Union and others. And this is why building the infrastructure for high-capacity rail needs to start right now: when the demand increases as population goes up, it will be able to be met, will reduce congestion and crowding.
Building rail for Seattle needs to be a forward-thinking process, because building tunnels, elevated sections and even creating at-grade right-of-ways takes lots of money and especially time. If we build now, we will be ready to accommodate the future need for transit.