Depends what the person has instead. I'm a British citizen but hold a different EU passport instead. I've had it since I was young, and when it came to renewal as an adult I didn't want to jump through silly hoops to get a British one like finding my parents' marriage certificate and being interviewed to check I'm not a terrorist, so I renewed my other passport instead (other bonus, it was slightly cheaper).
The things I have found are:
some places don't like foreign passports as ID (rare but true)
some official paperwork allows you to just write down your British passport number, rather than having to send in a copy of your foreign passport to verify that it's genuine. This happened on my student loan paperwork.
some places get confused by foreign passport numbers (mine is full of letters)
lots of questions about "why don't you have a British passport?"
benefits of British child passport: far more sturdy than my child passport, which was printed on stuff resembling green hand towel paper and confused everyone. Benefits of my adult passport: reinforced unlike those floppy British passports.
slightly different rules on whether I need a visa for certain countries, though a British passport wouldn't give me access to any more countries, just different ones (my passport is joint top with the UK for access). A couple we knew with different passports went on honeymoon together for a tour of multiple countries, and they needed to get visas for different places (e.g. one had a more complicated process to go to Russia).
I can go to my country's consulate abroad, but would also be able to go to the British consulate because I'm still a British citizen. Might be useful if one was shut and the other was open.
As you can see, since I have an EU passport plus British citizenship it doesn't make much difference, except for tiny little things.