बड़े (bade) is used in multiple settings while comparing ages, height, social status, property, prosperity, importance etc. here बड़े (bade) symbolizes big which literally means "big in context of age" and is a respectful adjective where age difference is not a big deal. if you use budhe then that means the bigger person is too old than the speaker. and this word is rather not used often when comparing ages but instead while describing ages as an adjective. also, budhe is a more disrespectful comparison for comparing age and is never used in a formal and respectful setting. also, when used as an adjective budhe means old and wise and sometimes if described by a young person to an old person then it could mean that the person is too old than what is acceptable. moreover, budhe is used to describe someone who is old, has lived for many years and is no longer young. badhe when used as an adjective describes a grown up, adult. so the sentence literally means that the persons are grown up and very big in age with comparison to the speaker/subject but are not a very old persons
Going Hindi to English, you would infer 'big' vs. 'old' from context.
I'm just another student like you, so take what's next with a grain of salt.
Going the other way around, I'd suggest using बूढ़ा to to say a person is 'old' (i.e. no comparison). It conveys respect in a way बड़ा doesn't seem to. Seeing बहुत बड़ा struck me as a bit less respectful.
बड़ा seems to be fine to use for people's age when used in a comparative sense, like 'my big brother', 'he is older than me', etc.
Above is based on what I've seen in Duolingo so far and in modest outside learning.
Side note: never use पुराना for people. It's only used for 'stuff'.