"When" vs. "what time"

The main difference is that the latter is really only answerable with some kind of wall-clock time answer, while the former can be more vague. This makes sense if you think about it. If the question is "What time...?" then the answer would have to include a time, right?

For example, if I ask a teammate, "When is our next game?" then "Next Tuesday" is a perfectly acceptable answer.

However, if I were to ask, "What time is our next game", then "Three PM." would be an acceptable answer (although he should probably include the "next Tuesday" part too, unless our games are usually on Tuesdays).

However, both of those phrases read as two sentences to me. If I were writing them I'd put a question mark after the word dear and a period at the end. As formulated, they both kind of look like the questioner is asking the other person if the questioner is hungry.

The first is grammatically sound. The interrogative adverb "when?" cooperates with the pronoun-verb phrase "we are meeting". You see this if you replace the "when?" with an adverb of time, such as "soon" (or a prepositional phrase, such as "at six"). Hence "we are meeting soon" makes sense as an answer (and so does "we are meeting at six"), replacing the interrogative adverb with a definite adverb (or prepositional phrase).

We are meeting soon.

We are meeting at six.

The second sentence is unsound - except when spoken informally - because "what time?" is an interrogative noun phrase, and cannot be replaced by an adverb. The noun phrase "what time" does not cooperate with the pronoun-verb phrase "we are meeting". You see this if you replace the "what time?" with any noun. The resulting answer makes no sense:

We are meeting day.

We are meeting six o'clock.

However, this way of putting it works when, instead of using "what time?", you use "at what time?" This turns the noun into a prepositional phrase, which, as we've seen, can be replaced with either a prepositional phrase or an adverb, because the two are grammatically equivalent.

We are meeting later.

We are meeting at six o'clock.