Summary
From preceding articles on this topic (the Four Signs):
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There are three levels of Exile: Physical, Mental and Divine.
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These Four Signs for the Coming of Moshiach:
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Increase in Informers
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Reduction in Torah Scholars
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Depletion of Funds
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Loss of Faith in the Redemption –
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They are corroding the spirit of the Jewish People, promoting Mental Exile.
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This hope is compared to an Old Dying Tree that's being attacked by strong winds, while its roots are decaying, and the surrounding soil becoming infertile.
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But Moshiach will then come, compared to a New Growth out of the Old Tree.
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He'll then rectify the Four Signs.
But what's wrong with the Old Tree, that it has to go?
Old Tree =
Hope for Physical Redemption
New Growth =
Hope for Mental & Divine Redemption
The New Growth is the
Offshoot & Root Sprout
coming out of the Old Tree:
וְיָצָ֥א חֹ֖טֶר מִגֶּ֣זַע יִשָׁ֑י וְנֵ֖צֶר מִשָּֽׁרָשָׁ֥יו יִפְרֶֽה
And a shoot shall spring forth from the stem of Jesse, and a sucker shall sprout from his roots.
(Isaiah 11:1)

A new tree growing out of trunk of the existing (Old) Tree
But sharing the same trunk and originating above the surface –
it is evidently a continuation of the Old Tree.
A renewal. Restoration.
This aspect of Moshiach comes to fix up the mess on the surface
(the Four Signs)
The problems we already know about.
The term "חטר" is synonymous with Resilience.
Moshiach needs to fight the "Wars of G-d" against negativity and worldly affairs, where strength and resilience are a must.
The "חטר" comes out first.
Followed by:
The Shoot / חטר

A new tree growing out of the roots of the existing (Old) Tree.
Looks like a new tree on the surface
(as opposed to the Offshoot).
It originates from a deeper, more concealed place
(directly from the roots).
This aspect of Moshiach is about what’s going on under the surface
(G-dliness – what we don’t see).
However it is now coming out to the surface
(about time!).
The term "נצר" is synonymous with weak growth.
G-dliness is not about physical force.
Yet it is this aspect that bears fruits (יפרה),
fruits symbolizing the goal of the tree – the Ultimate Redemption
(as the subsequent verses describe in detail).
The Root Sprout / נצר
NOTES & REFERENCES
Moshiach comes to revive faith and cause Repentance: Pri Tzadik on Genesis 1:2. Taamei Haminhagim p.439. Zohar III p.153b. Note Faith and Repentance are the antidotes of the Four Signs which stand for heresy.
Faith in the Redemption brings the Redemption: Midrash Mechilta Beshalach 14:31. Also note that Repentance (the result of the renewed faith) is a pre-condition for Redemption – Rambam, Laws of Repentance 7:2.
And a shoot shall spring forth from the stem of Jesse, and a sucker shall sprout from his roots: Isaiah 11:1. The relationship between this prophecy, describing Moshiach as a new tree (in the form of shoot and sucker), and the Four Signs which are compared (by the Maharal) to a tree being attacked – has already been explained.
חטר (Shoot) is a new tree growing out of trunk of an existing tree: MALBIM on this verse.
חטר is synonymous with Resilience: Metzudat Tzion and MALBIM on this verse.
נצר (Sucker) is a new tree growing out of the roots of an existing tree: MALBIM on this verse.
נצר is synonymous with weak growth: Associated with the word נוצר (Notzer) which means to guard, that is this aspect needs to be guarded – Ibid, This is possibly why this aspect has been under the ground. Also note נצר shares the same letters as רצון (Ratzon) which means "Will" – which, once revealed and asserted, can be most powerful.