Logic is a word that has come to be synonymous with reasoning, but its origin is that of referring to the field of studying human reasoning. Given that study (logos; -ology) is a social activity of human reasoning, the field itself is recursive, ie. it is reasoning about reasoning. It has come to replace 'reasonableness'. Nowadays, a person is more likely to be referred to as logical than reasonable. There is still a distinction in that the field of study is not singular, but social. An individual can reason, but cannot solely encompass the field of logic. The word "illogical" has no meaning as an opposition to the field of study as a whole, but has become a synonym for being "unreasonable." or that aspect studied within logic that is fallacious.
So it is understandable how you two may be using the word from different contexts. The field of study can and does encompass the study of fallacious reasoning and "illogical" arguments. In this roundabout manner, logic can thusly include the illogical and be referred to as illogical itself.
If that isn't terribly clear, another way to think about the distinction is that a person can reason, but cannot unreason. A person can be either reasonable or unreasonable though. The former is the individual's action, while the latter is a description from an outside observer. Likewise, logic is a field of study while there isn't really a thing referred to as illogic, but people can be either logical or illogical. The words unreason and illogic do exist, but as back formations of their adjectival forms.
In logic, there is a distinction between "validity" and "soundness." A valid argument is one that simply follows from the premises (the assumptions). Whereas a sound argument is one in which the premises have to also be true:
An
argument is
sound if and only if
1. The argument is
valid, and 2. All of its premises are
true.
For instance,
All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.
The argument is valid (because the conclusion is true based on the premises, that is, that the conclusion follows the premises) and since the premises are in fact true, the argument is sound.
The following argument is valid but not sound:
All organisms with wings can fly.
Penguins have wings.
Therefore, penguins can fly.
Since the first premise is actually false, the argument, though valid, is not sound.
When it comes to religious arguments, validity can be an aspect in that it should not contain any internal contradictions. The soundness of the arguments tend to be unverifiable though because those arguments attempt to contextualize all of life or known reality, so the contextualization (premises and assumptions) tends to be outside our boundaries.