The difference comes about because the original carol is a hymn about Mary. There are many different variations - the original had 23 verses in all (see here for the complete history on The Hymns and Carols of Christmas) - but they definitely fall into two camps: the Catholic Carol and the Protestant version.
Here is the Catholic Version as printed in the St Paul's Lessons and Carols program:
1. Es ist ein Ros entsprungen,
Aus einer Wurzel zart,
Wie uns die Alten sungen,
Von Jesse war die Art,
Und hat ein Blümlein bracht
Mitten im kalten Winter
Wohl zu der halben Nacht.
2. Das Röslein, das ich meine,
So uns das Blumlein bringt,
Maria ist's, die Reine,
die uns das Blüm'lein bracht';
Aus Gottes ew'gem Rat
Hat sie ein Kindlein g'boren
Bleibend ein reine Magd.
3. Das Blümelein, so kleine,
Das duftet uns so süß,
Mit seinem hellen Scheine
Vertreibt's die Finsternis.
Wahr' Mensch und wahrer Gott,
Hilft uns aus allen Leiden,
Rettet von Sünd' und Tod.
Translation:
1. A Rose has sprung up, from a tender root. As the old ones sang to us, its lineage was from Jesse. And it has brought forth a flower in the middle of the cold winter, right upon midnight.
2. The Rosebud that I mean, of which Isaiah told, is Mary, the pure, who brough us the Flower. At God's immortal word [strictly, "God's eternal Council"], she has borne a child remaining a pure maid.
3. The Flower, so small, that smells so sweet to us, with its clear light dispels the darkness. True man and true God! He helps us from all trouble, saves us from sin and death.
Note that in this version, the rosebush is the line of Jesse, Mary is the "Rosebud", and Jesus is the "little flower" that she brings forth.
The Choir sang that word for word except the second verse which followed a more Protestant understanding of the identity ofthe "Rose" in the song (which appears to stem from Praetorius 1609 according to Wikipedia):
2. Das Röslein, das ich meine, davon Jesaia sagt,
hat uns gebracht alleine Marie, die reine Magd.
Aus Gottes ew’gem Rat hat sie ein Kind geboren
wohl zu der halben Nacht.
Translation:
2. The Rosebud that I mean, of which Isaiah told, Mary alone, the pure Maid, has brought us; Out from God's eternal Council, she has borne a child half-way through the night.
So, you see, there is quite a difference. In Praetorius' version, the Rose is always and completely Jesus only, not Mary. My oldest German hymnals don't even have this carol in it - it appears only to have re-entered German Lutheran usage in the middle of the 20th Century. In one way, it just goes to prove that we often don't really understand the words of the carols that are so familiar to us.
