A perfect example, for me, is Hymn #470 in the Hymnal 1982. You can hear the usual tune, Beecher, here.
In seminary, we learned this old hymn with a new tune, St. Helena. It is Hymn #469 in the Hymnal 1982. All of a sudden, I heard the words as if for the first time! You can hear the St. Helena tune here.
Better yet, listen to it here performed by the Harvard University Choir.
These are the words - I hope you can hear them anew!
There’s a wideness in God’s mercy like the wideness of the sea;
there’s a kindness in his justice, which is more than liberty.
There is welcome for the sinner, and more graces for the good;
there is mercy with the Savior; there is healing in His blood.
There is no place where earth's sorrows are more felt than up in heaven;
there is no place where earth's failings have such kindly judgment given.
There is plentiful redemption in the blood that has been shed;
there is joy for all the members in the sorrows of the Head.
For the love of God is broader than the measure of the mind;
and the heart of the Eternal is most wonderfully kind.
If our love were but more faithful, e should take him at his word;
and our life would be thanksgiving for the goodness of the Lord.