Oh, much, much smaller. In fact, much smaller than turbines. You can get about 10Kw (that's 13.7hp) from 1kg of brushless motor. That's 10:1. Not a single gas turbine can even come close to those power-to-weight ratios. They do roughly 5:1 at best. And combustion engines do about 1:10, so they're really inefficient.
But as you hinted at, that's only half the story. When you add energy storage to this you presently add on a lot of weight. It's the energy storage that needs to be solved - the propulsion/prime mover is fine and already there. That's why the breakthrough will come through batteries or power cells and why that's so important to solve. Great progress has been made in this department (just think of how small and how long your phone keeps its charge compared to your first portable cell) but much more is needed before it can be a viable alternative. We'll get there though, and faster than people think.