Peaceably vs. Peacefully
Solution 1:
Webster's Dictionary of Synonyms (1942) groups peaceable and peaceful (along with irenic, pacifist, and pacifistic) under the entry for pacific:
Pacific, peaceable, peaceful, ... are sometimes confused because they all involve the idea of affording or promoting peace. But pacific applies chiefly to persons, utterances, acts, influences, ideas, and the like, that tend to make peace or to conciliate strife; [examples omitted]. Peaceable applies also to persons, their actions, words, etc., but it describes their character or quality as peace-loving, as disposed top avoid strife, or as inclined to keep peace, rather than their aims or tendencies; [examples omitted]. Peaceful applies especially to a life, a condition or state, a period or age, a country or people, in which peace prevails or there is no strife, but it may apply to anything which manifests peace, especially peace of mind, or provides an opportunity for such peace; [examples omitted].
Thus, from this dictionary's point of view, the main difference between the two words is that peaceably indicates in a manner likely to maintain peace and minimize strife, while peacefully suggests in a manner demonstrating a state of peace and an absence of conflict.
With regard to the specifically legal meaning of peaceable, Black's Law Dictionary, revised fourth edition (1968) has this:
PEACEABLE. Free from the character of force, violence, or trespass; as, a "peaceable entry" on lands. "Peaceable possession" of real estate is such as is acquiesced in by all other persons, including rival claimants, and not disturbed by any forcible attempt at ouster nor by adverse suits to recover the possession or the estate. [Citations omitted.]
Significantly, Black's does not include a legal definition of peacefully, from which we may deduce that peacefully is used far less often in statutes and ordinances than peaceably is. Often, English (and American) law will use fairly carefully defined terms across multiple jurisdictions in preference to very similar but less well-defined (in law) words. That appears to be what's going on with peaceably versus peacefully.
The key feature of peaceable possession or enjoyment of a thing appears to be possession or enjoyment without contention or dispute over many years. In the provision you refer to in your question, the rule has some elements in common with the law of adverse possession (namely, the openness and continuousness of the possession or enjoyment), and it serves somewhat similar ends (namely, not to upset a status quo that all interested parties have acquiesced in for many years).
Conclusion
Peaceable and peaceful have similar and largely overlapping meanings in everyday English, but they are not so completely interchangeable that a dictionary cannot distinguish different tendencies or colorations in their meaning.
But aside from the issue of whether the two terms mean different things in general usage, peaceable appears to be a much more distinctly defined and widely used term in the context of statutory law; and courts may have settled on a shared definition or interpretation of the terms "peaceable entry," "peaceable possession," and "peaceable enjoyment" that has no counterpart with regard to the terms "peaceful entry," "peaceful possession," and "peaceful enjoyment."
Solution 2:
Peaceable has a particular meaning in this instance.
Peaceable possession
A continuous possession of property that is uninterrupted by legal suits or other action intended to oust the possessor from the property. Peaceable possession does not preclude there being adverse claims against the possessor, so long as no actual attempt has been made to dispossess him or her.
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