Finch has been skipper since October 2014 but with Smith already captain of both the Test and one-day team, national selector Rod Marsh said it was the right time for him to now take over in the shorter format.
"Aaron Finch has done a very good job captaining Australia in T20 cricket. He will have benefited enormously from the leadership opportunity and will remain a highly-respected leader within the Australian squad," said Marsh.
"We think now is the right time for Steve to lead Australia in all three forms of the game as it offers us important continuity, not only ahead of the World T20, but beyond that tournament as well."
A key surprise in the 15-man squad was the inclusion of Nevill, who is yet to play limited-overs cricket for Australia, but he was preferred to Wade whose recent form with the gloves and bat prompted the change.
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"We want Australia's best wicket-keeper playing in this tournament and we consider Peter Nevill to be the best in the country right now."
Ashton Agar and Adam Zampa, both yet to play T20 cricket for Australia, claimed the spin spots ahead of Nathan Lyon and Cameron Boyce, while Josh Hazlewood spearheads the attack.
Finch (hamstring), Nathan Coulter-Nile (shoulder) and James Faulkner (hamstring) were all named in the squad subject to proving their fitness.
"Overall, we believe the squad we have selected is well-balanced and has the experience and talent required to be successful in India and help us win the World T20 title for the first time," said Marsh.
Australian squad: Usman Khawaja, David Warner, Shane Watson, Steve Smith (capt), Glenn Maxwell, Mitch Marsh, James Faulkner, John Hastings, Peter Nevill, Adam Zampa, Josh Hazlewood, Aaron Finch, Nathan Coulter-Nile, Ashton Agar, Andrew Tye.