I recently found out that in the UK, there is this card called the biometric resident permit (BRP), which I suppose functions like the visa sticker on the passport. Since I do not have a BRP, is it necessary to have one? What are the advantages of having a BRP rather than just a visa sticker on the passport?
The Biometric Residence card is not required in all circumstances, and I've found no significant advantage in having it. Note that I am a non-EU citizen, so your experience may be different.
The best source of information on the card that I've found is at https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/261497/brp-information-leaflet.pdf
Immigration authorities will tell you when you need to get one (it'll be detailed on the form you use), and there's no reason that I know of to get one sooner than that. This is detailed in that document on page 4 "Who has to apply for a biometric permit".
I was required to get one as part of my conversion from a "Spousal Visa" to "Indefinite Leave to Remain". Until then, my visa was stuck into my passport. The card has a few (minimal) advantages, according to the document above.
The main use is proof of identity for employers, or any place that you would need to prove your identity (such as at a bank, or when enrolling as a student). This may be useful if you don't have a card-base driver's license, or have lost your passport or similar.
If you have a biometric residence permit and are a foreign national, you are required to show it to a potential employer before they hire you (see the above document). Note that if you don't have the biometric permit, employers may still hire you, but they will require different evidence of your visa - such as the sticker in your passport.
You will be required to obtain a BRP in certain circumstances. According to Biometric residence permits:
You’ll have to apply for a biometric residence permit (BRP) as part of your application to extend your visa or settle in the UK.
You’ll also have to apply for one if you want to transfer your visa to a new passport or apply for certain Home Office travel documents.
That page doesn't specifically list any advantages of having one, other than the fact that it's required in a number of common long-stay circumstances.