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.
In general, never trust words in the English language to be phonetic! This is largely a consequence of English being such a fast-evolving language, and importantly, owing its vocabulary to many linguistic sources: Latin, Old French, Anglo-Saxon (a.k.a. Old English), Norse, and many others.
In this case it seems we have French to thank. This etymology is given online, and explains the supposed strange pronunciation:
debt
late 13c., dette, from O.Fr. dete, from L. debitum “thing owed”, neut. pp. of debere “to owe”, originally, “keep something away from someone”, from de- “away” (see de-) + habere “to have” (see habit). Restored spelling after c.1400.
In other words, debt comes via the Old French dete, which itself derives from classical Latin debitum. The b sound got lost due to French phonological rules/convention, and hence the French-origin pronunciation in English. Evidently, after the end of the Middle Ages in the 15th century, there was much revived interest in the classical world, and the spelling reverted to include the original b. Pronunciation, of course, stayed the same.
(Note that this sort of evolution occurred with many different English words, and occurred at the same time many new Latin words entered the English language.)
Best Answer
Wikipedia references linguist Edward Carney's (Senior Lecturer in Phonetics at the University of Manchester) A Survey of English Spelling in this explanation on silent letters, so the names of these various terms are not standardized. He distinguishes between two types of 'silent letter': auxiliary and dummy:
Auxiliary letters, paired with another letter, constitute digraphs which represent a single distinct sound. These auxiliary letters are further classified into exocentric digraphs—where the collective sound of the digraph is different from the individual sound of each of its two letters. The individual letters are rarely considered silent because each letter contributes to the overall sound of the digraph. There are two categories of exocentric digraph:
—and endocentric digraphs, where the sound of the digraph is the same as that of one of its letters, which are classified into three groups:
Other forms of silent letters:
consonant cluster silent letters: silent 〈th〉 in asthma
spurious silent letters: silent letters that are added to adopted words post factum to more accurately reflect even earlier origins, such as
Non-rhotic 〈r〉 accents: 〈r〉 is silent in such words as hard, feathered
h-dropping accents: 〈h〉 is silent
silent letters in compound words: compound words are often simplified in pronunciation, while their spelling remains the same. For example, cupboard and breakfast were once pronounced as written, but were then simplified over time.
The Wikipedia section goes more in-depth on the purpose of silent letters and how they originate in words.