Why is the enthalpy change not zero in an adiabatic process?
The Enthalpy $H$ is defined as $H=U+PV$. Therefore, 

$$\Delta H=\Delta U + P\Delta V +V\Delta P$$

For an adiabatic process, $q=0$. therefore from the first law of thermodynamics, 
$$\Delta U = q +w =q-P\Delta V$$
$$\Delta U=w=-P\Delta V$$
Substituting this in the first equation you get, 
$$\Delta H=V\Delta P$$
If $\Delta P$ is zero during the process (Isobaric) then the enthalpy change will indeed be zero, but it will be non-zero if the process is not isobaric.