Subsist derives from Latin literally meaning to stand still or to remain. Sub means "under", and sistere means "to stand" or "to place"; imagine placing a piece of paper under a weight so that it does not leave.
Subsist is generally used to indicate survival on a minimum of resources, but can also be used to indicate something continues (retaining a certain state) or even just exists. In your example, subsist seems to be used to indicate persistence. The "art of law" continues, or survives in the accumulated experience and expertise.
Note that subsist does not require a complement. You can simply say "The art of law subsists." In your example, the complement "in" was added to indicate where the art of law persists.
In contrast, consist derives from Latin literally meaning to stand together. Con- derives from the Latin cum, meaning "with" or "together," and again sistere means "to stand."
Consist is used to indicate either what a subject is composed of, or what it is comprised in.
In its modern usage, consist always requires a complement indicating either what the subject is composed of, or what the subject is included within.
Consist also has an older definition allowing it to indicate that something simply exists, but this is an obsolete use of the word.
From Webster's Revised Unabridged Dictionary (1913):
To Consist, Consist of, Consist in. The verb consist is employed chiefly for two purposes, which are marked and distinguished by the prepositions used. When we wish to indicate the parts which unite to compose a thing, we use of; as when we say, Macaulay's Miscellanies consist chiefly of articles which were first published in the Edinburgh Review." When we wish to indicate the true nature of a thing, or that on which it depends, we use in; as, There are some artists whose skill consists in a certain manner which they have affected." Our safety consists in a strict adherence to duty."
See also Dictionary.com:
Consist verb (used without object)
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to be made up or composed (usually followed by of):
"This cake consists mainly of sugar, flour, and butter."
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to be comprised or contained (usually followed by in):
"Her charm does not consist only in her beauty."
Best Answer
It was common in ME and EModE to introduce a clause serving as the object of a preposition with the complementizer that. This use dropped away in ModE.
You may find examples in OED I, s.v. That conj., definition I.1.c.. It's toward the bottom of the final column of the second page at the link.