I was always under impression that "most important" is correct usage when going through the list of things.
We need to pack socks, toothbrushes for the trip, but most important is to pack underwear.
However watching Apple presentations I've noticed they always say "most importantly".
Could you please explain how to use this correctly?
But most importantly, Gallo said, it's evident that Cook cares deeply about Apple.
While I agree with previous answers that the distinction in the sentences in the question has to do with adjective vs. adverb usage, I also think it's useful to note a related issue concerning the adverbial usage of the specific word importantly, especially in the phrases more importantly and most importantly.
For some reason, importantly was (probably unjustly) targeted by style guides and usage experts in the mid-20th century. After having encountered a couple overzealous older editors who insisted on rephrasing almost any use of the word importantly, I'm somewhat heartened to read answers and comments here by people who seem not to even realize this was once a significant editorial issue.
There's a bit of the history of the controversy here:
In 1968, “Winners and Sinners,” a periodic bulletin published by the New York Times, noted that, at the head of a sentence, “the adverbial phrase ‘more importantly’ modifies nothing in the sentence. What is wanted in constructions of this kind is ‘more important,’ an ellipsis of the phrase ‘what is more important.’”
Other authorities felt the same way, including Strunk and White’s “The Elements of Style,” which categorized the sentence-modifying “more importantly” as a misuse and told writers to avoid it by replacing it with “more important” or some other term.
Strangely, this flurry of opposition to “more importantly” was concentrated mainly in the 1960s and ‘70s as people got to thinking about the term and decided that it didn’t make sense.
Usage guides in the past couple of decades generally tend to downplay this issue or state that it was always an imaginary problem. Bryan Garner in his Modern American Usage notes that this logic simply can't apply to many other sentence adverbs, like notably or interestingly (e.g., one generally wouldn't begin a sentence, "Most notable,..." but rather "Most notably,..."). He goes on to say:
The criticism of more importantly and most importantly has always been rather muted and obscure, and today it has dwindled to something less than muted and obscure. So writers needn't fear any criticism for using the -ly forms; if they encounter any, it's easily dismissed as picayunish pedantry.
Despite Garner's judgment, I have still in recent years encountered people who appear to adhere to Strunk & White's unjustified statement that the word importantly should always be rephrased. (Actually, in this case, this arbitrary pronouncement seems to originate with White; the rule did not appear in the original Strunk.)
Thus, while there is a distinction to be made between the adjective most important and the adverb most importantly, writers should be aware that there are still those out there who think that even adverbial uses of most importantly are incorrect and should be changed to most important. (By the way, this modification is generally explained in older usage guides by some logic like: at the beginning of a sentence, what you should be saying is "What is most important..." and most important is an abbreviated version of that. Why the word important(ly) among sentence modifiers is singled out for this treatment is never explained.)
The difference is that "most important" is kind of like a noun, while most "importantly" is an adverb. They mean something on the following lines:
But most important (the most important thing/person) is Bill.
Most importantly (what is most important is that), Bob is dead
The second example, from Apple, reads as if it were missing quotation marks. I parse it thus: "Most importantly", Gallo said, "It shows that Cook cares deeply about Apple." I suspect the writer was not using a sentence adverb to convey that what Gallo said was most important; but rather, he was trying to say that Gallo himself said "most importantly, it shows..." (If it were a sentence adverb, why the comma after "said"?) Clarifying that subtlety, with proper punctuation, is more important than whether one uses "important" vs. "importantly" (but, of course, if it was in fact a quotation, it should have quoted Gallo using whatever word he actually said!)
To cite example 1 ("Most importantly [what is most important is that], Bob is dead") grammatically means that Bob is "importantly dead". Maybe that means Bob is a martyr or that Bob, though dead, has the willpower to be dead in a glorious fashion. The critical paraphrase, "what is most important is that", is the translation of the reader's assumption of what the writer meant without the writer caring what he meant.
This has certainly been answered many places, but here is a quick reference.
“Most important” is an adjective that can stand on its own in certain cases as the prefix ‘the’ and noun ‘thing’ can often be omitted. “Most importantly” is an adverb that happens to almost never be used in front of verbs, but rather before or after the subject-verb.
In your first example, there are three basic options:
The part after the "but" is turned into its own particle separate from "we need to" and most important is used as an adjective. ‘Thing’ can be omitted as implied and, if it is, ‘the’ may also be omitted (“...but most important is to pack underwear.”).
If we change it to a proper list with the word ‘and’ instead of ‘but’, ‘which’ appears to specify underwear as the most important thing in which case ‘the’ should not be omitted (“...and the most important which is underwear.”). However, a comma may be used instead as in “...and most important, underwear.”. You may see this last one without a comma, but that is incorrect as, without it, most important becomes an adjective for ‘underwear’ rather the the position it occupies in the list.
In this case, “most importantly” modifies “pack”. If it was taken out of the list and moved before, it would become "We need to, most importantly, pack underwear.” (commas optional) or "Most importantly, we need to pack underwear." Because “most importantly” is an adverb, it cannot modify ‘underwear’ and so there is no comma between them.
In your second sentence by Apple, the adverb, “most importantly”, is used as it modifies the verb “to be” conjugated as “is” in the contraction “it’s” (“...most importantly it's...”).
There are quite a few sites that explain this "ly" thing, which has become a phenomenon in just the past decade or so, before that we never heard "important(ly)" very often. The Grammar Girl site has an easy to understand explanation on using "ly," along with several others. IMHO, ly should be added with more and not most, and should never be at the end of a sentence, but that's just me, and of course it depends on the context. If it sounds awkward, it probably is incorrect.
I can't think of any way that "most importantly" can be used correctly. It is possible that there is a way to use it but it would be rare. I put it in the same realm as "firstly" and "secondly" which should never be used.
At issue is the ly, which some find unnecessary (and somewhat snooty). Many sticklers do not accept importantly in the two sentences that follow: I left my bed and, more importantly, I left the house. Most importantly, Churchill was a statesman. Critics of those sentences would prefer "more important" (what is more important, I left the house) and "most important" (what is most important, Churchill was a statesman).
Other experts declare the phrases acceptable with or without ly. But since brevity is a virtue, why not drop the ly and save yourself a superfluous syllable?
