Bäckerei vs. Backstube
While both Bäckerei and Backstube relate to bakeries, they have distinct meanings and uses in German.
Bäckerei
A1Bäckerei refers to a bakery shop where baked goods are sold to customers.
Ich kaufe jeden Morgen frische Brötchen in der Bäckerei.
(I buy fresh rolls at the bakery every morning.)
Die neue Bäckerei an der Ecke hat köstliche Kuchen.
(The new bakery on the corner has delicious cakes.)
Backstube
B1Backstube specifically refers to the kitchen or room where baking takes place, often not visible to customers.
In der Backstube wird der Teig vorbereitet und gebacken.
(In the bakery kitchen, the dough is prepared and baked.)
Der Bäcker verbringt die meiste Zeit in der Backstube.
(The baker spends most of his time in the baking room.)
Summary
In essence, Bäckerei is the public-facing bakery shop, while Backstube is the behind-the-scenes baking area. A Bäckerei typically contains a Backstube, but a Backstube can exist without a Bäckerei, such as in large-scale commercial baking operations.