How are aspirated letters different from silent letters when pronouncing a word?
How are they similar?
Aspiration = a strong burst of air that accompanies the pronunciation of certain sounds (in English, voiceless stops in onset position in stressed syllables or word-initially). This also occurs with the [h] consonant. In either case, a burst of air is produced.
Silence = no sound at all.
Now, just because a word is spelled with an "h", this does not mean that there is aspiration; whether or not there is aspiration is wholly dependent on the pronunciation. Words such as "hour" have no aspiration (in any dialect I am familiar with) — this means that there is nothing there, and the "h" is just representing some former historical pronunciation. On the other hand, "historical" has aspiration (the "h") in my dialect of English, but not in other dialects.
So, if you perceive silence where there is an "h" in spelling, it is not aspiration. Aspiration is audible/detectible in a spectrogram.
English is a language with a million glottal stops. Every word that is written as starting with a vowel either starts with an aspiration, a glide, or a glottal stop. For example, the word "once" starts with a glide, the word "I" starts with a glottal stop, the word "union" with a glide, and the word "it" is lightly aspirated when spoken south of the Dixie line. Other words spelled with aspirations do not or sometimes do not have an aspiration. "Herb" goes either way, but "hour", "honor" and "honesty" do not. Unstressed aspirations, for example, "it", "historical", are treated as if unaspirated even when aspirated, for example, "an historical romance", but not "an hotel" if the emphasis is on the first syllable: "a hotel" when pronounced "a HOE tell", but "an hotel" when pronounced "anno TELL" when emphasis is on the second syllable and the aspirated syllable is unstressed. Where I am from, the first syllable is stressed when an adjective, "a hotel restaurant", but when a noun, the second syllable is stressed, "an hotel by the sea." Similarly, although both words are aspirated, it's "an historical treatise", but "a history of the United States". People who pronounce words without the aspiration in their dialect are perfectly grammatical to use "an" in place of "a". These examples are the pronunciations in my class and area, not models.
When two vowels come together, they are separated with the glottal stop when there is no aspiration, for example, "the hour is nine". "Hour" begins with a very pronounced glottal stop, where the air is unmoving because of a blockage in the throat for a full beat. It sounds like a cough and is notated with a backquote in Hawa`ian, as Hawa`i is not pronounced hah-why, but hah-wah-`ee, with a stop between the A and I. English speakers mispronounce it with a glide, that is, hah-wah-yee. Certain of our vowels end in glides, like long I and long O, so combining them brings out the glide onto the next syllable, like "Ohio" = "Oh-hy-yo", or "buoy" = "Boo-wee", but never across words. Certain foreign accents are immediately recognisable by the inability to pronounce the glottal stop, which is hard to say, because there is no glyph for it, a lot of languages don't have one, and most English speakers are unaware that they are beginning most sentences with this consonant. Similarly, native English speakers give away their language backround immediately by using it between words which end and then begin with a vowel, for example, "Maria Elena", pronounced "MAR-ee-ah-LENN-nah", not "Mar-ee-uh ` Ell-AY-nah", as Anglos say it, with a consonant inserted between the vowels to keep them apart. We hear it quite glaringly as foreign when people say "I yam what I yam", having omitted the glottal stop and carrying the implied glide onto the next word, a violation of the rules.
So, in short, it is a very complicated matter, this business of aspiration, and it depends strongly on dialect. In Yiddish areas of NYC, "I yam" is perfectly normal. Down South, it sounds horrifyingly foreign/Yankee. I never realised that I spoke the word "it" with an aspiration until I tried to use IBM ViaVoice to enter the handwritten first draft of my first novel. I had to use search-and-replace to turn all of the "hit"s into "it"s. Of course, the word "hit" has an especially hard aspiration in Southern English, to distinguish it. We tell them apart just fine, but Yankees complain of us constantly "hitting", "Hit's a nice day, hisn't tit" is what they hear, or they call us breathy. ViaVoice assumes you are from Murray Hill, New Jersey, and upper class. I am prep-school/Ivy League educated, but my accent was unintelligible to the computer, or in Boston, Mass. I couldn't even order a bagel the first time I went there, my Magnolia Mouth was so bad. And I'm not even from the deep South, only from Washington DC. And if you grow up there, you assume everyone talks as if from Washington. It's the capital, after all, and the model for the rest of the nation, and news people reporting from Washington sound just like us.
Born a snob, I have been working on ridding myself of it. No dialect is superior to the others, and even the most educated accent can be unintelligible in a place where it is never heard.
Speaking of glottals, in the Mountain West, the word mountain is pronounced with a glottal stop in place of the T, and the N is swallowed. When they say Mow-en, they are not wrong. Adele or Lily Allen, when they drop internal Ts for the glottal stop, are not mispronouncing anything, they are being true to their voice. So it drives me crazy when people assume I am stupid because I have a Southern accent, although I used that fact to get out of a ticket in LA once, saying, gee whiz, out in New Mexico all the speed limits are 75 for highways, with my best Magnolia Mouth. A look of contempt and no ticket. Does not work in the MD-DC-VA area. I open my mouth and they hear Preppie, and I have to be careful not to sound like I am talking down to the cop. In England, sounding Posh is not a complement, and people like John Major, who sound perfectly normal but British to us Americans, are speaking risibly because of their attempt to be posh without getting the shibboleths right.
So I say don't worry about it if you are a native speaker of English, and if you are learning, try to learn the English of a particular place, and then you will be consistant with yourself.
I believe that whether it is treated as aspirated or silent frequently deals with the domain of snobbery. The snobbish prefer to treat most h's as silent, so that they have the opportunity to write "an hotel" or "an hospital," when clearly these are not the standard pronunciations. It falls into the same category as the misuse of the verb "graduate" (e.g. "he graduated college" rather than "he graduated from" or "was graduated from", etc.