Is the Old Testament God different from the New Testament God?

No.
God is and always has been the exact same. The God that watches over us now is the same that created the universe, died on the cross, and rose from the dead.
Some people have said that the Old Testament had an angry God while the New Testament had a loving God. This is 100% untrue.
The same God that destroyed the cities of Sodom and Gomorrah is who John speaks of as being “love”.
God is unchanging.
God is love.

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Psalm 102:25-27 — "Of old You founded the earth, And the heavens are the work of Your hands.
"Even they will perish, but You endure; And all of them will wear out like a garment; Like clothing You will change them and they will be changed.
"But You are the same, And Your years will not come to an end.

Hebrews 1:10-12 — And, "YOU, LORD, IN THE BEGINNING LAID THE FOUNDATION OF THE EARTH, AND THE HEAVENS ARE THE WORKS OF YOUR HANDS;
THEY WILL PERISH, BUT YOU REMAIN; AND THEY ALL WILL BECOME OLD LIKE A GARMENT,
AND LIKE A MANTLE YOU WILL ROLL THEM UP; LIKE A GARMENT THEY WILL ALSO BE CHANGED. BUT YOU ARE THE SAME, AND YOUR YEARS WILL NOT COME TO AN END."

Hebrews 13:8 — Jesus Christ {is} the same yesterday and today and forever.

John 3:16 — "For God so loved the world, that He gave His only begotten Son, that whoever believes in Him shall not perish, but have eternal life.

Romans 5:8 — But God demonstrates His own love toward us, in that while we were yet sinners, Christ died for us.

1 John 4:8 — The one who does not love does not know God, for God is love.