Am I misunderstanding what JSON-LD and JSON Schema are?
About JSON-LD, yes; In JSON-LD document, JSON-LD is defined as:
a JSON-based format to serialize Linked Data
Can these two even be compared?
It would be better to compare JSON-LD with JSON (not JSON Schema). Or you could compare JSON Schema with the other encoding syntax schemas e.g. XML Schema.
Note: The following section is about the remaining questions (considering the difference between JSON-LD and JSON).
Choosing between JSON and JSON-LD depends on the situation and context of use. Generally, JSON is a markup language in the syntactic level where the data encoded is just machine-readable. But JSON-LD is being used to semantically markup the data, to make them become not only machine-readable, but also machine-understanable by providing additional syntax to JSON for serialization of Linked Data.
Recommended resource to understand the detailed differences between JSON-LD and JSON is the JSON-LD document published by W3C.