When an airport has helipads, why would a helicopter take-off or land on a runway?
I know you asked specifically about runway versus helipad departures, but I thought I would lay out all the options available to the heli pilot, and you can see what all the choices are.

There are three different areas on an airport that a helicopter can operate from:


Non-movement area -- this is the area of the airport that is not controlled by the ground or tower controller. It often consists of aircraft parking areas or an area leased by a tenant.  It is separated from a movement area by this the marking below. Helipads may be found in non-movement areas, such as the Coast Guard helipads at SFO and JFK.  
Movement area -- this is the area of the airport controlled by a ground controller, and separated from the runway area by this marking: . Helipads can be found in movement areas.
Runway area -- this is the area controlled by the tower controller, and contains the runways used for landing and departing traffic.


This picture from KHWD shows the three areas:



Regardless of which area is used, almost heli departures from towered airports are coordinated directly with the tower controller because the aircraft enters the controllers airspace.

At non-towered airports, helis operate out of the flow of fixed-wing traffic, and need to self-announce like any other aircraft.

Additionally, helicopters can make three different kinds of departures:


max performance take off -- which begins with a near-vertical segment, and then transitions to a horizontal. This is typically used from a confined area, and looks something like this: 
normal takeoff -- this begins in an in-ground effect hover with a horizontal segment until translation lift is assured, then transitions into a climb: 
running takeoff -- when there is insufficient power to begin with an in-ground effect hover, then the helicopter may begin its departure by sliding (skids) or rolling (wheels) along the ground, until translational lift can be obtained, then transitioning into a normal takeoff: 


The choice of departure area and departure type are for the most part independent, but they are obviously linked based on performance and obstacles.

The departure area is an operational decision. When the helicopter is operating from a remote part of an airport, the pilot will often make a departure from a non-movement area (helipad or ramp) so she doesn't have to taxi to the runway.  If the helicopter is a passenger flight, such as air-taxi, it will most likely operate from the runway or a helipad in the movement area.

The departure type is a performance decision, based on obstacles, weather, weight, and available power. Hot &amp; high departures might require a running take-off. Lightly loaded twin-engine copters in normal weather can safely make high-performance departures over obstacles in the desired direction of flight.