Although the two quantities are often confused, ΔH and q are completely different values. First, definitions:
q is the amount of heat transfered to the system. It is one of the two means of energy transfer during most processes studied in thermodynamics. The other means of transferring energy is through work. Since these are the only means by which we can transfer energy between the system and the surroundings, we can write the change of internal energy of our thermodynamic system as:
ΔE= q + w
(note: some books denote internal energy as U)
Enthalpy (H), on the other hand, is a thermodynamic potential, much like internal energy (E). It is essentially a measure of the potential energy of the system. Enthalpy is defined by the following equation:
H = E + PV
ΔH = ΔE + Δ(PV)
By adding in the definition of ΔE, we can see that
ΔH = q + w + Δ(PV)
Why do ΔH and q often get confused. Because, lets consider what happens in a system at constant pressure. Because pressure is constant Δ(PV) = PΔV. Also, for a process occurring at constant pressure, w = -PΔV. So, our equation for the change in enthalpy simplifies to:
ΔH = q (valid for constant pressure ONLY!)
Since most chemical reactions occur at constant pressure (i.e. in open flasks exposed to atmospheric pressure), ΔH is very useful to chemists because it readily relates an easily measured quantity (heat) to a thermodynamic potential (enthalpy). Of course, once you start looking at transformations that occur under variable pressure, calculating ΔH is no longer very simple.
Now, there is an important fundamental difference between enthalpy and heat. Lets say you are studying a process that takes your system from P1, V1 and T1 to P2, V2 and T2. Now, there are many different paths one can take between the two thermodynamic states. Along each of these paths, the values of q and w will differ. However, no matter what path you take, ΔH for the transformation will always be the same because ΔH depends only on the initial and final states of the transformation.