Yes, they will know. For the most part, they will not care, with one important caveat.
They will care only if you have a non-smartphone plan and you plug a smartphone into it. Even if you block data transmission, they will add a smartphone data plan to your account and charge you for it if you put a smartphone (which they identify based on the IMEI, or unique identifier, for the phone) on their network. If you go back to a "dumb" phone, you can always have that data plan removed.
If you already have a smartphone/data plan you are happy with, you can change phones all you want by simply moving the SIM from phone to phone. If they see it happen more than a few times a month, they may (or may not) question you about it.