It’s not quite as simple as “choral part” vs. “voice type,” however. While they are often treated as the same range, alto as a choral part refers to the part above the tenor part. But the contralto voice type will often sing in the tenor range, rather than above it. It makes sense: the prefix “contra” tends to mean “lower than” in music terms, e.g. contrabass instruments being lower than bass instruments.
A key reason for this is that the vocal types are divided into 3, while the choral parts are most often split into 2 or 4. There are the sopranos, mezzo-sopranos (often just mezzos), and the contraltos. But choral parts are usually soprano and alto, with each sometimes subdivided, giving Soprano 1, Soprano 2, Alto 1, Alto 2. Even when I do see three-part female choral pieces, they are usually SSA, i.e. 2 soprano parts and one alto part. (I honestly didn’t even know mezzos existed until I got to college.)
So you wind up with contralto voices that sing Alto parts (orAlto 2), but also sometimes some mezzo-soprano voices (or Alto 1). And you’ll have both mezzos and sopranos that sing Soprano 2 parts, or just the lower Soprano parts.
Something similar happens with male parts, as well. The baritone can usually cover the classical choral bass range. But a the bass vocalist usually has an extension below this, at least for a few notes. And the baritone can often cover the lower parts of the tenor range, albeit with a heavier production. So you still have the same mix with two/four choral parts, but 3 vocal types. The main difference is just that the baritone has only been acknowledged as a separate part more recently, and is still often just considered a subtype of bass. A baritone may start out singing lighter bass parts and then transition to singing heavier tenor parts. But their voice range didn’t change so much as they changed focus, and better developed their upper range.
SATB hasn’t always even meant men and women. The choirs were originally all men, with the sopranos being unchanged male voices, either using children or the castrati, whose name probably tells you what that means. Thus it’s not surprising that such doesn’t map perfectly to the vocal classification types of men and women.
So, while the “vocal part vs. vocalist” is a good first order approximation for alto vs. contralto, I’d say the bigger difference is between how the voices are divided. The contralto vocalist can usually sing lower than most alto parts. It’s not uncommon, especially in smaller or less advanced choirs, for a contralto to cover tenor parts. (My grandmother did this a lot in church choirs.)