Pull Force of an electromagnet How do I calculate the pull force of a cylindrical electromagnet to Iron as a function of distance? 
Is there any difference betweeen magnets and electromagnets? 
 A: To quote the K&J Magnetics calculator page:

Most online calculators determine pull force based on a theoretical calculation of the flux density. With a few assumptions, flux density (in Gauss) can be related to the expected pull force. Unfortunately, this simplification often fails to match experimentally measured data.

Theoretically, you compute the magnetization that the magnet induces in the iron plate and compute $dE/dx$, the change in the interaction energy between the permanent magnet and the iron plate with increasing separation distance, in order to find the pull force.
What they seem to be saying is that their calculator uses empirical methods instead, based on a history of discrepancies between experiment and theory.  This is not implausible; ferro-magnetism is quite complex, and discrete magnetic domain size and hysteresis (among other effects) can prevent simple versions of theory from matching experiment.
It is worth noting that if you enter a separation distance of 0 inches in the calculator that the pull force for the iron plate configurations and the magnet-magnet separation force are the same.  This is because the magnet essentially creates a copy of its magnetic field configuration in the iron plate.  This, of course, assumes a thick iron plate (compared to the thickness of the magnet), and no longer works when there is a gap between the plate and the magnet.
In order to calculate the pull force as a function of distance, you have to compute a consistent magnetic field solution.  In other words, you have to solve for the magnetic field configuration that exists when a magnet is a given distance from the iron plate.  You can compute the interaction energy of the magnetized iron with the permanent magnet as a function of minute changes in distance, and use the relation
\begin{equation}
  F=\frac{\partial E}{\partial x}
\end{equation}
to solve for the pull force.  Solving for the self-consistent magnetic field configuration is not trivial, as any student of E&M will tell you.  J.D. Jackson addresses this subject in his canonical text book, Classical Electrodynamics.  In chapter 5, $\S 5.9$, he discusses the equations necessary in order to solve the boundary value problem.  This treatment does not address the microscopic vs. macroscopic nature of magnetic fields and only briefly touches on the subject of hysteresis in $\S 5.11$.  However, these are the equations that one would use to compute the field due to a permanent magnet at a given distance from a "virgin" piece of iron.
A: It's a very hard problem.
If magnetism behaved like gravity, you could break up the plane and magnet into small dipoles and sum up all of the dipole-dipole interaction energies for all possible pairs (this would make a hex-tuple integral!). The force is simply the rate of change of the total energy. However, you need to first calculate the induced magnetization of iron, which is complicated and has a "saturation" behavior. The paper:
https://web.archive.org/web/20150601014125/http://www.gris.uni-tuebingen.de/people/staff/spabst/magnets/Magnets_in_Motion.html
Lays out the needed calculations. Of course, the cylindrical symmetry will reduce the calculations needed.
