Is there a difference between anti-anxiety and anti-depression drugs? [PREFACE- I've already asked this on Health.SE, but I've drawn a blank there. The place is practically deserted. Also, this is NOT a personal medical question; I'm simply trying to understand the difference between the two classes of drugs.]

I've recently come across two terms "Anxiolytic drugs" and "Anti-depressant drugs". I wanted to know how they differ from each other; if they do.

I mean, sure, "anxiety" isn't the same as "depression" but do the effects of these drugs differ widely?

For example [And this is just an example];

Chlorodiazepoxide is classified as an anxiolytic drug, while Phenelzine is classified as an anti-depressant. Say I had a patient A who's been prescribed Chlorodiazepoxide; patient B has been prescribed Phenelzine by a different physician. Barring possible allergic reactions, or some kind of known sensitivity to either drug, could they possibly take each other's medication and hope for more or less, the same effect? As in; A takes Phenelzine, but B takes Chlorodiazepoxide. [I just cooked up that example, only to see the difference between the two classes of drugs]

To make myself clear, I reiterate:

I would like to know if the two classes of drugs (anti-anxiety and anti-depression) are the same or not. If not, mentioning how the chemistry behind the action of each of the two types of drugs differ would be appreciated.
[Basic]


Receptor: a molecule in a cell membrane that responds specifically to a neurotransmitter, hormone, antigen or other substance.
Agonist: Substance which initiates a physiological response when combined with a receptor.
Antagonist: Substance which inhibits the physiological response when combined with a receptor.
Allosteric: Substance which alters the activity of an enzyme by means of a conformational change induced by a different molecule.


[Response]


GABA receptors are related with anxiety [1]. Anxiety occurs because of low concentration of GABA in the brain. Therefore, Chordiazepoxide is an allosteric agonist.
Dopamine receptors are related with depression [2]. Depression occurs because of low concentration of Dopamine in the brain. Therefore, Phenelzine is a non-selective and irreversible monoamine oxidase inhibitor, which means that the brain does not recycle/degrade the monoamine neurotransmitters (Which dopamine is one of them).