Starkiller Base is more advanced
It can destroy multiple planets simultaneously

Notice how the beam splits to destroy the entire Hosnian system.
From the script for The Force Awakens:
EXT. REPUBLIC SYSTEM - NIGHT
At a massive distance we see PLANETS- THE REPUBLIC CAPITAL
SYSTEM -- INCREASINGLY LIT, THEN POWERFULLY HIT BY THE VAST
FIREBLAST WHICH OBLITERATES IT ALL!
Poe confirms that the Starkiller Base destroyed the entire system:
POE:
Finn's familiar with the weapon that
destroyed the Hosnian system. He
worked on the base.
The Death Star had a single superlaser, and as such would be hard-pressed to destroy multiple planets in a single shot—after which, of course, it had a significant recharge time of 24 hours (according to Star Wars: Everything You Need to Know).
Starkiller Base can hit targets at greater range.
According to Finn:
"It fires through a hole in the continuum that it makes itself.
Everybody was calling it 'sub'-hyperspace."
—The Force Awakens (novelization)
In addition,
Assuming that the rotation and inclination of the planet had been
taken into account, the released blast of concentrated phantom energy
would travel along a perfectly linear path, punching a small Big Rip
through hyperspace itself until it left the galaxy— —or encountered
something in its path that was of sufficient mass to intercept it.
—The Force Awakens (novelization)
Finn nodded. “General Hux told us it’s the most powerful weapon ever
built. He said that it can reach halfway across the galaxy.” Fresh
murmurs of disbelief greeted this latest assertion. “And in real time.
Because it doesn’t reach across the galaxy; it reaches through it.” He
shook his head, which was starting to hurt from the effort of trying
to explain what he had overheard but did not understand.
—The Force Awakens (novelization)
Note that Finn's audience reacts to the revelation of the weapon's range with disbelief. This certainly seems to imply that the range is greater than anyone had believed possible, which must have included the range of the Death Star.
In addition, this last quote shows that Starkiller is instantaneous
The Death Star's weapon, as a laser traveling through real space, would travel at the speed of light. The Star Wars galaxy is over 100,000 light years wide.
Leia knew perfectly well that the Rebellion wasn’t coming to Corva
sector anytime soon, not with the fate of the galaxy to be decided
nearly a hundred thousand light-years away. The people of Sesid were
alone—just as the people of Basteel were.
—Moving Target: A Princess Leia Adventure
Even if it could reach halfway across the galaxy, it would take 50,000 years to reach its target—more than enough time for ships with hyperdrives to evacuate the entire planet.
Planetary shields may have also improved.
It is also quite possible that planetary shields have gotten stronger since the era of A New Hope, forcing the development of larger superweapons to penetrate them. Thus, even if the Starkiller base might not have been necessary to destroy Alderaan in the past, it would be necessary to destroy the Hosnian capital as of The Force Awakens
And lest you think that the Death Star is superior, because it can actually move around...
Like the Death Star, the Starkiller Base is mobile
