Q:

How can I say "How many more apples does John have than Phil?" in Spanish?

Does the following translation make sense in Spanish? What would be the appropriate translation?
English:

How many more apples does John have than Phil?

Spanish:

¿Cuántos más manzanas tiene John que Phil?

A:

Firstly, "cuánto/cuánta/cuántos/cuántas" is an adjetivo interrogativo which, as all adjectives, must agree with the noun it qualifies — "manzanas", in this case.
Secondly, the adverb "más", if paired with some kind of quantifier, is generally placed after the noun:

¿Cuántas manzanas más necesitas?
  Necesito tres manzanas más.

Then I wouldn't separate "más" and "que" because that makes it more difficult to parse.
Summing up, I'd say:

¿Cuántas manzanas más que Phil tiene Ali?

You can also say:

¿Ali tiene cuántas manzanas más que Phil?

The last one sounds clunkier to me, but it could sound natural if pronounced with a different intonation. I would pronounce the first one with a falling intonation, and the last one with a rising intonation (or a "less falling" one), as though I wanted the addressee to complete my sentence.
Of course you could go with a different wording:

¿Cuántas manzanas más tiene Alí con respecto a Phil?

I don't love the last one, but that is personal preference.

A:

Your attempt is correct. You just missed the fact that "manzanas" is feminine, so "cuántos" has to match the gender and be "cuántas":

¿Cuántas más manzanas tiene Ali que Phil?

However, to me it sounds a bit better to swap both terms and say:

¿Cuántas manzanas más tiene Ali que Phil?

