The word we want here is "instance", which refers to an example or occurrence, not "instant" which is a very short period of time.
2. Article use:
There is more than one way for courses to differ, so we shouldn't use the definite pronoun ("the"). It would better to use "a" or "one" instead.
The items being compared need to be clear and comparable. In this case, we want to compare different kinds of courses, not a course and a classroom, so the wording needs to make this clear.
3. Word form:
"Correspondents" is the plural form of the noun. What we need is the adjective, "correspondence". Note also that "correspondent's" or "correspondents'" would be correct from a grammatical point of view, but would mean something quite different.
We could use either the singular ("course") or the plural ("courses"), but the items need to be comparable: "correspondence course/classroom course" or "correspondence courses/classroom courses".
5. Word confusion:
Do not confuse "then", which refers to what happens next, with "than", which we use to make comparisons. But here, we need another word entirely: "from". One thing can be "different than" another, but it "differs from" another.
6. Word form:
The word "equipment" modifies the noun "cost". We could use the adjective form ("equipment cost") or we could make it possessive ("the equipment's cost"). But we can't make it plural ("equipments"), so we cannot use the plural possessive ("the equipments' cost").
7. Verb use:
You do not need the auxiliary ("are") with the modal ("ought"). You can say "ought to have", but simplicity is usually better, so consider using the word "need" instead.
8. Number error:
There are several students, so there should be more than one “computer”.
In this sentence, "for" is being used as a conjunction (meaning "because"). But it sounds like a preposition ("expensive for students"), so it would be clearer if we used a different word ("because", "since", or "as").
10. Agreement error:
The word "another" is singular (literally, "an" other). But we need the plural form: "other".
A semicolon is used between independent clauses when there is no coordinating conjunction. We should not use it after a phrase like "for instance". Use a comma instead.
You should not use a comma to separate the subject from its verb, so you should omit the comma before "is".
You should also hyphenate compound adjectives like "up-to-date".
"Modern" means "up-to-date" so you don't need both words.
For instance, one way in which correspondence courses differ from classroom courses is that equipment costs are very expensive, as students need to have computers, internet, and other up-to-date things.