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.
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.