Both values are correct, but each figure is only giving you part of the total losses. You have to add the two values.
The equation you posted delta Pressure = ( Density/2 ) * ( v2^2 - v1^2 ) is based on
Bernoullis equation , which is true for a
non viscous liquid, ie
no friction losses: perfect world.
The app (I suspect) is calculating
only the friction losses, and for this you need another equation such as
Hazen Williams.
To illustrate, if you have a 4.12m high column of water in a tube, tank or whatever and you have a hole in the bottom, then in an ideal frictionless world the water will be forced out of this nozzle at 9m/s. (Bernoulli)
If you connect a pipe to this hole, then in the real world friction will slow the water down. To get back up to 9m/s, you will have to make the water column higher by adding another 0.5m (in this case).
So you now need 4.62m for 9m/s.