I'm discounting Crowder from the list as he's been in the league for 4 years. So has Zeller and Sullinger. In '06-'07 Perk had 4 years under his belt. So looking at the real comparison:
2006-07 Celtics: Tony Allen, Ryan Gomes, Gerald Green, Al Jefferson, Leon Powe, Rajon Rondo, Sebastian Telfair, Delonte West
2015-16 Celtics: Kelly Olynyk, Marcus Smart, James Young, Terry Rozier, RJ Hunter, Jordan Mickey
Jefferson kills Olynyk. That's not even close.
You'd have to give Allen even odds with Smart (though I like Smart more)
Young, Rozier, Hunter and Mickey have potential but then again so did Rondo, West, Green, Gomes and Powe.
So the difference is Big Al and that's a big difference. Big Al was considered perhaps the best up-and-coming young big man in the game. He was a 22 year old who had just finished putting up 16/11 in his first real heavy action. His splits continued to show growth as well as he put up 20/11 in the last month. He was anticipated to be a star (and until he blew out his knee he was). There isn't anyone on the current roster who has demonstrated that much potential on the court.
Thus '06-07 gets the nod with the 3 years and under statement and truthfully it's not all that close. What's different is the mid-level talent. The '06-'07 team had PP. This team has IT, Crowder, Bradley (Sully and Turner as well but their contracts make them less valuable); all of whom are either just entering their prime or well within it; all whom are exceptionally cheap.
When you add those in, plus Brad, plus the picks it's quite obvious that this team is in a better situation - both currently and with the potential for future growth - than that team was.