How do we reconcile Deuteronomy 22:5 with John 7:24? To do so, we should consider the context of the NT quotation. My answer is that they do not need to be reconciled because Jesus is not discussing the issue of proper dress in John 7. He is discussing the issue of exceptions to the law against working on the Sabbath.
Moses gave you circumcision (it is, of course, not from Moses, but
from the patriarchs), and you circumcise a man on the sabbath. 23 If a
man receives circumcision on the sabbath in order that the law of
Moses may not be broken, are you angry with me because I healed a
man’s whole body on the sabbath? 24 Do not judge by appearances, but
judge with right judgement.’
The issue was this: when, if ever, is it permitted to "break the Sabbath." Or, more properly: "are there certain principles that supersede Sabbath observance." Jesus made the point that, according to Jewish tradition, circumcisions can be performed on the Sabbath. Although it is not mentioned in the Torah itself, this was apparently the case in Jesus' day, and remained so throughout Jewish history. According to Maimonides:
When a circumcision takes place at the regular time (on the eighth
day), it supersedes the prohibition of work on the Sabbath. But if it
is to be performed after the regular time, it neither overrides the
obligation of the Sabbath nor of the festivals. (Mishneh Torah,
Circumcision 1)
In this sense asking us to reconcile Dt. 22:5 and John 7:24 is a bit like asking us to reconcile "apples" and "oranges" -- because Jesus was not making a pronouncement about proper dress; he was pronouncing with regard to healing/circumcision on the Sabbath. When Jesus said not to judge by "appearances," Jesus was not referring to manners of dress; he was referring to the letter of the Law as opposed to the spirit of the Law, especially as regards Sabbath observance.