Translation:The young girls are lying on the grass.
Calling these suffixes "cases" is unnecessarily scary. Cases are for Indo-European languages, and are far more complicate than these, but Hungarian belongs to another family.
In Hungarian, PREpositions are attached after the words, instead of before, therefore they could be called POST-POSITIONS (or "postposition suffixes"). That is all.
Using Latin expressions to describe these post-positions just makes thing sound more difficult. It also feels a bit weird to me, since Hungarian is completely unrelated to Latin.
"Inessive" comes from "in esse", Latin for "to be in". "Superessive" comes from "super esse", Latin for "to be on". "Adessive" comes from "ad esse", Latin for "to be next to".
Well, you've got to call them something in English. I suppose Hungarian has its own terminology not based on Latin.
Regarding the difference between postpositions and (postpositional) suffixes: Is it just a spelling convention that distinguishes one from the other? That is, the former are written as separate words, whereas the latter are attached as suffixes. In the spoken language, there would not seem to be much difference between the two.
Or is there some basic grammatical difference? If not, why is a distinction made?
We agree on that: the only difference between postpositions and postposition suffixes is that the suffixes are attached (I confirmed with my wife, who is Hungarian).
My point is: there is a basic grammatical difference between Hungarian postposition suffixes and cases in Indo-European languages. My knowledge of case-based languages is limited to Latin, ancient Greek, German and a bit of Czech. May I ask you about Swedish? It should be close to German.
There is one more consequence to the difference between suffixes (attached) and postpositions (separate): suffixes generally come in different flavours and have to follow the vowel harmony of the parent word. Postpositions always stay the same.
And there's no need (it's even wrong) to call them "postpositional suffixes". Suffixes always attach to the end of the word. Postpositions come after.
Hungarian has the same option; you can say kis lány (or write it as a compound word, kislány) and it corresponds just about exactly to "little girl" in English. Since that corresponds so well to "little girl", I would usually prefer to translate fiatal lány as directly as possible, with "young girl".