Why would ATC issue a takeoff clearance and also tell the same aircraft to maintain visual separation at the same time?
Yes, the Tower's separation standards are different if the following aircraft accepts the responsibility to maintain separation visually.  If the following aircraft can't (or declines the clearance to) maintain separation visually, the controller has to wait significantly longer between subsequent departures.

Typically, the transmissions from Tower are something along the lines of:


Tower: "Airline 123, line up and wait 25R, confirm A320 on departure roll in sight."
Pilot: Airline 123, line up and wait 25R, Airbus in sight.


After the A320 has rotated:


Tower: Airline 123, 25R cleared for takeoff, maintain visual separation on preceding traffic, they will be turning north.
Pilot: Airline 123, cleared for takeoff 25R, maintain visual separation on the Airbus.


The advisory to the crew about they will be turning north gives them the awareness that they won't be following the other aircraft for miles and miles, just until it reaches the point a few miles from the airport where some traffic turns north and other traffic turns to the south (or proceeds straight-out).  The aircraft will generally be sequenced by Ground Control so that they line up at the runway in a sequence such that the departure directions are staggered.  When this doesn't work, at PHX anyway, that tends to be when there is a break in the flow of departures for a minute or so, and during that time aircraft that have landed on 25L (in this case) and are waiting down-field to cross the departure runway, can cross and then taxi to the terminal.  Tower will often let the "waiting to cross" queue build up to three or four aircraft -- each holding short of 25R at a different taxiway -- while they launch departures rapidly, and then one departing aircraft remains "lined up &amp; waiting" while the arriving aircraft are cleared in rapid sequence across the runway.

And after they're all clear, the departures resume.

All in all, it's a pretty efficient way to "move the metal" &amp; get lots of aircraft into the sky as quickly as possible on clear days -- which PHX has plenty of!