Here are the lyrics to "Amazing Grace":
Amazing Grace, how sweet the sound,
That saved a wretch like me.
I once was lost but now am found,
Was blind, but now I see.
T'was Grace that taught my heart to fear.
And Grace, my fears relieved.
How precious did that Grace appear
The hour I first believed.
Through many dangers, toils and snares
I have already come;
'Tis Grace that brought me safe thus far
and Grace will lead me home.
The Lord has promised good to me.
His word my hope secures.
He will my shield and portion be,
As long as life endures.
Yea, when this flesh and heart shall fail,
And mortal life shall cease,
I shall possess within the veil,
A life of joy and peace.
When we've been here ten thousand years
Bright shining as the sun.
We've no less days to sing God's praise
Than when we've first begun.
Here are the lyrics to "By Grace I'm Saved":
By grace I'm saved, grace free and boundless;
My soul, believe and doubt it not.
Why stagger at this word of promise?
Hath Scripture ever falsehood taught?
Nay; then this word must true remain;
By grace thou, too, shalt heav'n obtain.
By grace! None dare lay claim to merit;
Our works and conduct have no worth.
God in His love sent our Redeemer,
Christ Jesus, to this sinful earth;
His death did for our sins atone,
And we are saved by grace alone.
By grace! Oh, mark this word of promise
When thou art by thy sins opprest,
When Satan plagues thy troubled conscience,
And when thy heart is seeking rest.
What reason cannot comprehend
God by His grace to thee doth send.
By grace God's Son, our only Savior,
Came down to earth to bear our sin.
Was it because of thine own merit
That Jesus died thy soul to win?
Nay, it was grace, and grace alone,
That brought Him from His heavenly throne.
By grace! This ground of faith is certain;
So long as God is true, it stands.
What saints have penned by inspiration,
What in His Word our God commands,
What our whole faith must rest upon,
Is Grace alone, grace in His Son.
By grace to timid hearts that tremble,
In tribulation's furnace tried,--
By grace, despite all fear and trouble,
The Father's heart is open wide.
Where could I help and strength secure
If grace were not my anchor sure?
Both hyms are about grace but understand grace in very different ways. "Amazing Grace" was written by John Newton and was about his own conversion experience. He was the captain of a slave ship and on his way back home he encountered a dangerous storm. He called out to God for mercy, God spared him, and he looked back on this as his conversion. The hymn was a retelling of how God's grace kept him from death and converted him and now he can look forward to singing about God's grace in heaven. It's a very subjective hymn and based on an experience unique to John Newton--not an experience shared by every Christian.
"By Grace I'm Saved" by Christian L. Scheidt is much different. It speaks to the reality of every Christian. All are saved by God's free grace. The author of the hymn does not appeal to his experience but points us to the Scriptures. We can know that God is gracious because they Scriptures say that He is gracious and the Scriptures are trustworthy and God is trustworthy. The hymn says that we are not saved by our own works but by Christ's atoning death.