Electric motors are 90% efficient, compared to combustions 25% (or jet's even lower efficiency), so as you can see one can't just compare energy density between batteries and fuel. Battery power never needs to become as energy dense as gasoline - only 1/4 of that - to compete. If you add the much higher power-to-weight ratio of electric motors (about 10hp/kg - blows away anything else, even jets) the gap closes even more.
But ultimately, what will make the revolution in electric propulsion is price. Oil can't be produced in your backyard, so they have you over a barrel (pun intended). The very definition of a captive audience. They'll squeeze you dry, and governments will tax it until the last drop, so sooner or later the cost benefits of electric will become irresistible, even to the point where you might accept a slightly lower speed/range/whatever. Electric can be produced in your backyard and they can't control it in the way the do with oil.
My predictions I still stand by - in 20 years time there will be very little gas powered cars sold. Electric will be the prime mover for almost all transportation in the future, for sure. Like Elon Musk said - "With the one exception of rockets - they will be fossil fuelled still". I agree.