For me, "да" as a conjunction sounds like a looser variant, "and [well] [maybe] also". A very slight additional meaning of something free/natural/careless, when used for a whole sentence.
"и" is a stricter "and", "definitely and", "inclusively", almost "both" (but less certain than the double "и... и...")

I'm not sure if this is how other people feel it. Technically it's also equivalent to the Ukrainian та/чи and is heavily used in folklore in natural enumerations (noun+noun; adverb+adverb), for example, Иван да Марья, дед да баба, вокруг да около и т.д.

There's also да which is an interjection, it's used as part of specific idioms.