Which one consumes more bandwidth: YouTube app or web browser?
You will want to confirm with your carrier, but it won’t matter if you connect via browser or via the app.
Remember, the owner of any network knows what IP address, thus what customer, connects to which server and how much data is transferred to/from that connection.  This is just looking at it from the most basic perspective and not taking into account that there are tools that can detect the type of content being transferred.
They know that your IP is connected to a YouTube server regardless if you use the app or browser and won’t throttle/count the data from those servers against your monthly allotment.
The reason I recommend contacting your carrier is because they may have an agreement with YouTube that stipulates using the app for the unlimited data (i.e. you can’t block adds or better tracking, etc.). So technically, it doesn’t matter; they know how much YouTube data you consume; it may be a contractual (business) issue that requires you to use the app.