The biggest difference is that case is a grammatical category whereas adpositions are a syntactic category, i.e. cases are properties across (certain kinds of) words, and adpositions are one type of words. The case of a noun/pronoun/noun phrase/determiner phrase (hereafter just "noun") is a property of that noun; different languages mark cases differently, some languages don't even mark case. Adpositions are a category of words that typically express temporal/spatial relations.
For example, in the English sentence I won the argument with her, the pronoun I is in the nominative/subjective case, and the pronoun her is in the oblique case. In English, the rule is that subjects have nominative case and complements of prepositions have oblique case. The noun phrase/determiner phrase the argument could have accusative/objective case or not, depending on your view of English syntax. You may say that all English nouns have case, but case is only realized in pronouns, or you may say that in English only pronouns have case, not nouns.
While in English, case marking is part of the noun but adpositions are separate from their complements, in some languages it is difficult to tell case marking and adpositions apart. For example, here are some Japanese sentences.
1. Taro-ga pan-o tabeta.
Taro-NOM bread-ACC ate
'Taro ate the bread'.
2. Taro-kara Hanako-e hanataba-ga okurareta.
Taro-FROM Hanako-TO bouquets-NOM were.sent
'Bouquets were sent by Taro to Hanako.'
In Japanese, case is marked by the so-called particles (ga and o in (1-2)), which have the same distribution as the adpositions (kara and e in (2)). Some people consider both of them as case markers, and distinguish the two as syntactic vs. semantic case markers (Kishimoto 2018, 448). However, adpositional phrases may be case-marked in Japanese:
3. Taro-kara-no Hanako-e-no tegami
Taro-FROM-GEN Hanako-TO-GEN letters
'letters from Taro to Hanako'
where the particle no marks the genitive case on the adpositional phrases taro-kara 'from Taro' and hanako-e 'to Hanako', making it difficult to distinguish between case-marking particles and adpositional particles on distributional grounds.
Kishimoto, Hideki. "Case Marking." Handbook of Japanese Syntax. De Gruyter Mouton, 2018.