In the context of Apple, I would agree that departments is the better word. HR is a department, as is Support.
Generally speaking, an entity can comprise several organizations, but in that case I would call that entity itself an organization (or, in case the entity is commercial, corporation). As a simple example, the UN is an organization, but so are its agencies such as the UNTSO and the WHO.
Looking at the Merriam-Webster definitions, I see that organization is an "association, society", while department is "a major division of a business". HR, Support, and iOS Development are certainly major divisions of Apple, but they can't be labeled as societies or associations in their own right.
I think that there is a very obvious answer to this question. The opposite of "better" is "worse." The opposite of "more" is "less." So, which of these sentences sounds correct:
I like dogs worse than cats
I like dogs less than cats?
The answer is clearly "I like dogs less than cats."
The sentence "I like dogs worse than cats" would be flagged by any native speaker as being incorrect. No one would ever say that. The answer, in my honest opinion, lies in semantics, not grammar.
Think about it deeply; it simply doesn't make sense to say that you like something better. The word "better" tells you something about the worth of something. Whereas the word "Less" tells you something about a quantity (less butter) or about the perceived strength of something (less light). To increase or decrease the force of your 'liking," only two appropriate words are available to you: you like something more, or less. You can't like something better or worse, because these are words that comment on the value of liking, not its intensity.
To say "I like cats better than I like dogs" means, semantically, that your liking for dogs is somehow better than your liking for cats. And it's obvious that that is not what most people intend to say when they use "better" in this context.
What they mean is that their liking for dogs is GREATER than their liking for cats, not that their liking for dogs is in some odd way BETTER than their liking of cats. Dogs may be better than cats, but our liking of dogs can't be better than our liking of cats.
Of course someone might think that it is better to like dogs than it is to like cats. But there are no "better" ways of liking something.
Best Answer
High is indeed the collocation of choice here. Big or large don't sound idiomatic. Both the Corpus of Contemporary American English (COCA) and the British National Corpus (BNC) confirm this.
The opposite, in fact, is low aspirations.
For the sake of completeness, here are the top 50 collocations from both corpora:
As you can see, most of these have to deal with kind rather than size. The ones that are of interest to you are, in this order, high, higher, highest, lofty, low, great, lower, deepest, growing for COCA, and higher, high, highest, lofty for BNC. So the verdict is pretty clear.