Learn Italian: Il mio quartiere non è così grande come il centro. - My neighborhood is not as big as the city center.

Così... come is the comparative of equality, meaning "as... as". You use it to show two things are at the same level. Combined with non here, it means the neighborhood is not at the same level of bigness as the center. If you used più... di, it would mean "more... than".

Could I say tanto grande quanto il centro instead?

Yes! When comparing adjectives like grande, così... come and tanto... quanto mean exactly the same thing. You can use either pair interchangeably in this sentence to mean "as big as".

Why is there an il before mio quartiere?

In Italian, possessive words usually need the definite article right before them, so it translates literally to "the my neighborhood". The main exception is for singular family members (like mio padre), but since a neighborhood is a place, it needs the il.

If the subject was feminine, would grande change to "granda"?

No, grande is an adjective that ends in -e in its singular form. These adjectives are exactly the same for both masculine and feminine nouns. They only change their ending to -i when they become plural.