Basically, じゃないですか (or informally じゃないか? , じゃない?, or じゃん?) can be used in a similar fashion to a rising tone ね/でしょう/だろう (asking for confirmation).
[negative form] んですか meanwhile is asking for an explanation about the action in question. This can naturally sound accusatory or part of some rhetoric. Think “Did I not tell you to pick up milk yesterday!?” or “Is it not true that our political opponents have only maintained power through bribery and corruption?”
So in the first example, 夕食を食べないんですか has a ring of “Why are you not eating dinner” while 夕食を食べるんじゃないですか has a ring of “Weren’t you planning on eating dinner?”
Also, note we have a grammar question thread, which is more likely to be responded to.
The difference is that ないんですか literally means ‘not?’ whereas んじゃないですか literally means ‘is it not?’ ない is traditionally considered an auxiliary verb in Japanese grammar, but how I’d see it (from an English speaker’s perspective) is that the ない version gets combined with the verb stem to form the negative form of the verb, whereas じゃない is a word on its own. It’s important to know what each component means. Speaking of which, ん is a contracted の, and creates an explanatory nuance and emphasis when combined with a form of だ・である（だ、じゃない、です…). It always comes after the dictionary form of the verb.
I’ll translate your sentence pairs a little differently, if perhaps less naturally:
夕食を食べ ないんですか 。
Won’t you eat/aren’t you eating dinner?
夕食を食べる んじゃないですか 。
You’ll eat dinner, won’t you?
この車は新しく ないんですか 。
Isn’t this car new?
この車は新しい んじゃないですか 。
This car is new, isn’t it?
あれは犬じゃ ないんですか 。
Isn’t that a dog?
あれは犬な んじゃないですか 。
That’s a dog, isn’t it?
Another example, for your reference:
Didn’t I tell you?/I told you so!
If we were to stick to my original translation pattern, that ought to be ‘I told you, didn’t I?’, but I felt that might make it a little less impactful. My point is though, んじゃない（ですか）is something like a generic question tag in English, whereas the [verb stem]ない form is more like a question asked using the negative. The difference in nuance, I believe, is the same as the difference between these two question types in English. One form implies a certain expectation by negating it, while the other states that expectation in full before raising the possibilty that it is false. That is all, as far as I know.