Usually with synonyms, the words have different levels of formality (words that come from French are usually more formal, and words from Old English are informal). For example, 'maternity' and 'motherhood' are synonyms, but 'maternity' (from French, maternité) sounds more formal and is used in scientific situations (maternal mortality rate) and fashion (maternity clothing), while motherhood (from Old English, moderhode) sounds less formal and is used in regular conversation. So, 'annoy' is not a very formal word, but it is more formal than 'to bug' which is slang. ('Annoy' comes from French, ennuyer; and 'bug' comes from Old English, bugge). You could use 'annoy' in a formal essay or in conversation, but 'bug' is only for informal conversation. This comes naturally to native English speakers, but I think you can learn with practice and by listening to when native speakers choose certain words over others.

Hi Ethan. Those two words have exactly the same meaning in context. The difference is register. Although both are medium register, “annoy” is a two-syllable word from French, while “bug” is a one-syllable word from Old English. “Bug” is thus just a bit more informal than “annoy.” Also, “bug” has additional meanings. As a verb, it can refer to electronic eavesdropping. It also retains its original meaning as a noun referring to insects and similar annoying little creatures. Additionally, there is the derived meaning of “software bug.”

In complete agreement with you. Another synonym for Bug is error. In case of a software. Bug can also be a device that is hidden and is used to record conversations (telephonic) without their knowledge.

if you're confused to choose the right word in right position Thesaurus of a dictionary can help you. the other tool I'm using in this case and helped me a lot was oxford collocation dictionary and Longman Activator too.