To start I'll need to list some pertinent definitions:
Office: a special duty, charge, or position conferred by an exercise of governmental authority and for a public purpose : a position of authority to exercise a public function and to receive whatever emoluments may belong to it (m-w.com)
Order: a group of people united in a formal way OR a special duty, charge, or position conferred by an exercise of governmental authority and for a public purpose : a position of authority to exercise a public function and to receive whatever emoluments may belong to it (m-w.com)
Yes, there's a big difference that makes the promise that the Aaronic Priesthood will never again be taken from the earth a big deal (see D&C 13:1), and it's that the Aaronic Priesthood can be shared with any worthy male, even outside the House of Israel.
I know the above image is hard to see, but if you were to zoom in it shows the House of Israel setup where male involvement equals patrilineal priesthood and the female role centers on producing heirs to the covenant. Make note that to be a Levitical priest you didn't just have to be male, you had to be a Jewish woman's firstborn, have no blemish, and your priestly father must bless you before he dies. Later on the angels come down from heaven to take Aaron and his sons through the washing and anointing process so that the priesthood can be extended to all worthy literal sons of Aaron (Exodus 40:12). This, in essence, becomes Levitical Priesthood 2.0. When the Early Church is formed, all worthy males are given access to the Aaronic Priesthood as we know it now, but this practice is eventually killed off (literally and figuratively). The Restoration brings back the more accessible Aaronic Priesthood.
As the above relates to keys, I see it this way. The apostles of the Church each hold 3 keys, which are 1) The Gathering of Israel, 2) the Abrahamic Covenant, and 3) Sealing power. Gathering of Israel is inherently female (you need to be born of a Jewish mother). Abrahamic Covenant is inherently male (your father must be a priest to receive His power). The sealing keys unite male and female for eternity. It's a nice triad. Note too that the office of Bishop rightfully belongs to a son of Aaron, and the gentile in that position is really just a seat warmer (see D&C 107:73). The keys to the Aaronic priesthood are for the outward ordinances (e.g. repentance, baptism), as well as ministering angels. The latter makes sense because the Aaronic Priesthood was initiated by actual angels descending to earth to minister to Aaron and his sons when the lower law (i.e. The Ten Commandments) were given.
Now let's talk Melchizedek Priesthood. It holds the keys to the spiritual things of the Church. You need it to confirm newly baptized members, to perform the sealing ordinance, and much more. We learn in D&C 107:4 that this priesthood is actually the Lord's priesthood, but to avoid overuse of his name it was called after Melchizedek who "was such a great high priest" (see D&C 107:4 & 2).
Let's dive into this a bit more, though...
Because Melchizedek was notably NOT from the House of Israel.
Yeah, say that again. Melchizedek was NOT from the House of Israel. He was not an heir to the blessing of Abraham, Isaac, and Jacob. How do we know this? Because Abraham paid tithes to Melchizedek as a respected elder, meaning he was alive at the same time and not a descendant of Abraham. You know who else was outside of the House of Israel? All gentiles, and everyone else born before Abraham... Which includes Father Adam himself!
I point this out because it makes the Melchizedek Priesthood standout and gives hope to people who feel stifled within the four walls of the House of Israel (I'm looking at you, ladies). We then step back and see a priesthood that centers on Melchizedek, but begins with Adam and will ultimately end with Christ's return. This is some true Alpha - Omega power going on. It follows that the associated covenant would be new (Adam) and everlasting (Christ), and encompassing of all promises and blessings in and out of Abraham's posterity (i.e. House of Israel and beyond).
We learn in Moses 5&6 that Adam did not always have this great priesthood. It wasn't until he gave sacrifices that he was overcome by the Holy Ghost and baptized and confirmed by Him through the Spirit. (That fact, by the way, underscores why the Melchizedek Priesthood pertains to Spiritual matters.) From that time, all the sons of Adam had a birthright, but only the chosen few received his patriarchal blessing to join this holy order. (This is the same pattern for Abraham: sacrificing in similitude of the Father and Son, subsequent covenants and power from father to son.)
At the same time Adam entered into a priesthood covenant with the Lord, Eve also had the Holy Ghost come upon her (see Moses 5:11). Like Sarah, there was likely a key associated to the female component of this priesthood power. We don't know much about this except that trees are an important source of power and imagery relating to women (see this breakdown, and this one re: wisdom, James 1:5 and the Sacred Grove) and could be the token for priestess-hood. When we later read about Deborah the Prophetess we might think she's an anomaly in the House of Israel (i.e. a woman with power beyond procreation), but the more I study this I believe she was calling down a birthright as a Daughter of Eve (not Sarah), which was symbolized by her standing under a (date palm) tree.
Why have I dragged you down this rabbit hole? Why are we here in the priesthood weeds? For a couple of reasons. 1. We need to appreciate why priesthood restoration is a big deal for the rest of mankind and 2. methinks that the next chapter for women and priesthood power will come by filling in the details of the New and Everlasting Covenant. Whereas our greatest promise had been eternal power onto our husbands in the House of Israel (i.e. Aaronic, Levitical) and temporal authority in the offices thereof, the door is opening on the Melchizedek side for greater things in store. --Likely alongside the Adamic order. The Melchizedek key that relates to this is, "the key of the mysteries of the knowledge of God." (See D&C 84:19)
Predictions --because no pre-General Conference post is good without predictions-- we're going to take bigger steps toward gathering Israel, and in parallel the New and Everlasting Covenant will endow more women with more power in the temple... or perhaps in a grove of trees! (Why not, right?)
For what it's worth, I ended up in this doctrinal pursuit based on the prompts and clues President Nelson sent. So there's that. :)
Follow-up thought: The temple exists to take the sons of Aaron and make them the sons of Adam again. The temple exists to take the daughters of Eve and make them the daughters of Sarah, as well.
