#Preface
First, it can be difficult to differentiate the two diseases even in the modern day, so I'll largely focus on them together [1, pg. 39].
Second, the majority of my sources focus on Europe, unfortunately, but I'll add in historical context where possible. If there's anyone with a better understanding of the history of depression and anxiety (or mental illness more broadly) in Asia or the Americas, I'd be truly fascinated to learn about it.
Third, if you want to read more on this subject, there's a lot of interesting history after the Medieval era. I think what happens to depression, anxiety, and mental illness in the Renaissance and beyond is one of the most fascinating phenomenon in the entirety of medicine's history. Still, as requested, I'll focus on the Medieval era in my answer.
Fourth, please be warned that the answer below will contain explicit reference to anxiety, depression, and suicide.
#Really Important Prefatory Note
Now, a fifth prefatory note (which, admittedly, makes me worry about the length of this comment). However, it also one which is exceedingly important when studying the history of disease, and especially that of mental illness. Please do not read on without first reading this note.
Illness is a socially constructed phenomenon. Bacteria, viruses, and cancers can cause symptoms, but "but relatives, friends, and health professionals surrounding a sick person classify and interpret those symptoms to determine if he or she is ill" [2]. What and how illness is classified varies from place to place and time to time. See the prescription of leeching and lobotomy for but two examples.
This makes the scholarship of illness challenging. "How a problem is defined affects how (or even if) society responds to the problem," and affects how historians analyze these problems just the same [3]. I am not going to wrestle with the definitions of "depression" and "anxiety" but you can and should. Or at least, keep in mind that this answer is based on 20th/21st century historians' understandings of the two mental illnesses.^1
#Medical
First, note that in the medieval era (and ondeed, for much of history) the term depression would not have been used. When writing about it as an illness, the term melencholia was common, deriving from the ancient Greek for "a mental disorder involving prolonged fear and depression." [3] It was not until the late 18th century that Depression became the standard in psychiatry. One psychiatrist, Adolf Meyer, argued "eliminating the term melancholia, which implied a knowledge of something that we did not possess ... If, instead of melancholia, we applied the term depression to the whole class, it would designate in an unassuming way exactly what was meant by the common use of the term melancholia" [qtd. in 3]. I will write on melancholia, as that is what the scholars of the time would use, but note that the term swallowed many diseases.
A lot of medieval understanding of depression is rooted in Roman scholarship (which itself stems from Greek scholarship). We can thus look at Rufus of Ephesus, who's 2nd century writings "became more or less standard in Islamic and Western medical writings for nearly fifteen hundred years." Rufus characterized those suffering from melencholia as "loomy, sad, and fearful, and he stated that the chief signs were fear and doubting." He said that "some are anxious about loud noises, and some wish they were dead, and some delight in washing, and some had an aversion to any food or drink, or detested some kind of animal; and some believed that they had swallowed snakes and the like." In terms of diagnosis, Rufus identified 3 categories of melencholia: "(1) a type in which “the whole body is full of a melancholy blood”; (2) a type in which “only the brain has been invaded”; and (3) a type in which “the hypochondria” were “primarily affected.”" Galen, another 2nd century author, elaborated on Rufus and declared "that the most essential symptoms were contained in the Hippocratic statement to the effect that prolonged fear and sadness constituted melancholia." Rufus and Galen would be major factors on the depiction of Melancholia in the Medieval era. ^2
The early etiology and pathogenesis of melancholia, like many diseases, was believed to be rooted in the humours. There's quite a bit of scholarship on humoural theory, so I wont go in detail here, but refer to this comment and the subsequent threads if you are unfamiliar with it or need a brush-up. In essence, the body was thought to have four humors -- blood, phlegm, yellow bile, and black bile -- and an imbalance in the humors was the cause of disease. Melencholia in particular was caused by an overabundance of black bile. This was thought to be caused by issues in the spleen, whose function (as described by Galen) was to filter "the thick, earthy, atrabilious humors that formed in the liver".^3
Alongside humoural theory, and well after its demise, was the system of the six things non-natural. These were usually "(1) air, (2)exercise and rest, (3) sleep and wakefulness, (4) food and drink, (5) excretion and retention of superfluities, and (6) the passions or perturbations of the soul" [3]. This "doctrine, within its category of the passions or perturbations of the soul or mind, provided a way to take account of the role of the emotions, including grief and sorrow, in the etiology of disease" [3]. The two explanations combined would last well through the Medieval era.
Treatment of melancholia was largely rooted around restoring equilibria, either that of black bile or that of the things non-natural. To reduce the excess of black bile, blood-letting might be prescribed, or purgation via "dodder of thyme, aloe, colocynth, or black hellebore" [3]. "Corrections for the six things non-natural were more varied, they ranged from "warm and moist" air to "pleasant company," and included "protections against suicidal inclinations" [3]. In particular, "Galen emphasized treatment with frequent baths and a moist, nourishing diet as sufficient to cure melancholia when it was recognized in its early stage" [3].
#Religious^4 ^5
I begun with the Romans above, but here I will move straight to the Medieval era.^6 First, on a broader note, the popular conception that many Medieval illnesses were blamed on some religious factor, whether demon or sin, is inaccurate. This is true even when focusing on mental illness, Kroll and Bachrach did a very interesting (albeit with acknowledged methodological limitations) study (sadly not focused on depression or anxiety) where they looked at primary sources (Saints' Lives, various other chronicles) and contrary to some earlier scholarship, found that "no evidence in [their] sources that there was significant reliance upon a belief that sin was the major causative factor of mental illness." Where it was cited, in all cases they identified a "strong propoganda motive" [4].
But on to depression. Prior to the deadly sin of sloth, we would have discussed the sin of acedia. Intriguingly, the origins of this particular and peculiar term are actually quite clear: the writings of Evagrius Ponticus, a 4th century monk living as "part of a cluster of hermit colonies [...] some forty to sixty miles south of Alexandria. He described it as a demon, which leads the monk to "hatred against this place, against life itself, and against the work of his hands, and makes him think he has lost the love among his brethren and there is none to comfort him." This is pre-medieval, but as above, it forms the foundation for acedia, and will change little through later centuries. Note before we move on that acedia is not the same as depression (or even melancholia) and that it is not the same as sloth. "Lassitude, weariness, inaction, carelessness, and neglect were all aspects of acedia," but it would be an inaccurate depiction to merely leave it there [5].
Evagrius' description would be conveyed into the Medieval era by John Cassian's *Twelve Books on the Institutes of the Coenobia [=monastaries] and the Remedies for the Eight Principal Faults." One of these faults was the spirit of accidie, to which Cassian devotes book 10. When it "has taken possession of some unhappy soul," he writes, "it [...] makes the man lazy and sluggish [...] he complains that he is cut off from spiritual gain, and is of no use in the place." When and if it overcame a monk, "it either [made] him stay in his cell idle and lazy, without making any spiritual progress, or it [drove] him out from thence and [made] him restless and a wanderer, and indolent in the matter of all kinds of work" [6].
Over the course of the Medieval era, acedia evolved. Some lists of the chief vices omitted it entirely; Gregory the Great subsumed it under dejection (he also lowered the number of sins to seven.) As the 11th century bled into the 12th, acedia underwent "a shift in emphasis from mode of behavior to state of mind." At this point, acedia was still largely restricted to men of God, but after the fourth Lateran council (of 1215) mandated mass religious education, acedia became more and more a part of public consciousness [5].
While acedia was a sin, it was still something that could be treated, by penance or confession. There was (and is?) a long history of priests as healers of the soul, and this medicalization of sin certainly existed in thase case of acedia. It was also following the Lateran reforms that acedia began its transformation into sloth: while it had previously been emphasized as a state of mind, "the common man's image of acedia came to center around spiritual idleness or neglect in the performance of spiritual duties. [5]"
Look to the next comment for the exciting conclusion to this post (and for the footnotes!)