I need to change my Biometric residence Permit (BRP) to my married name, however my passport with my maiden name is still valid (I renewed only last year). Can I apply for the change of name in the BRP and keep the passport with my maiden name?
No. UKVI will not normally allow you to have different names on your passport and BRP. If you want to change your name on your BRP, you must change the name on your passport first (indeed, if you change the name on your passport you must report this to UKVI).
You will need to complete form MCC.*
The fact that you cannot have different names on the two doesn't seem to be officially stated anywhere, but I know from personal experience that it is true. You could try filling in the form without changing the name on your passport, but I suspect that UKVI will ask to see a passport in your new name.
* Confusingly, the guidance at the top of the form says that if your "current grant of leave was made by means of a BRP" you should not use this form, but should "make an application for a new BRP, using the form for no time limit (NTL) or an application for transfer of condition (TOC)". This makes no sense, and I suspect it is supposed to say "current grant of leave was not made by means of a BRP".
I thought that statement about being fined 1000 pounds was terribly explained. I've been married for over a year now and my name hasn't changed since I got married. The marriage cert just shows the names of two individuals becoming married in the eyes of the law/courts. Nothing more, nothing less. As others have said, you may be fined if you don't let UKVI know within three months after you have LEGALLY changed your name.
I use my husbands surname with non-legal stuff. I'd like to change my name, but I'm not sure I want to go through that whole process. My husband is fine with whatever decision I make. I changed my name in my first marriage, then back to maiden name. Neither one was difficult, but name changes follow you for life, which I hate because it's a reminder that I got married when in my gut I knew I shouldn't have before I did. Anyway, not only is it passport, it's BRP, SS card, driver license, taxes and who knows what else. I guess the biggest reason I want to change my name legally is because if we have children, I want all our names to be the same (like on the birth certificate). May be silly, but it is just how I feel.