序(つい)でにWhile you are at it, On the occasion

Both express simultaneous actions, but differ in nuance. ついでに implies convenience, e.g. '買い物のついでに郵便局に寄った' ('I stopped at the post office while shopping'). がてら implies doing something while moving, e.g. '散歩がてら郵便を出した' ('I mailed a letter while taking a walk'). They also differ grammatically: がてら connects directly to nouns, while ついでに requires の.

ついでに and の間に can both express the idea of 'while', but are used in different contexts. ついでに implies taking advantage of one action to do another, e.g. 買い物のついでに銀行に寄った ('I stopped by the bank while shopping.') の間に, by contrast, means 'during/while (A), (B)' without the added nuance of 'while you're at it' that comes with ついでに.

Both ついでに and つつ involve performing multiple actions together, but differ in nuance. ついでに expresses performing a second action while it's convenient or appropriate, e.g. '買い物のついでに郵便局に寄った' ('I stopped at the post office while shopping'). つつ is a formal expression indicating simultaneous actions or states, e.g. 'レシピを見つつ料理をする' ('cook while looking at a recipe').

Both ついでに and ～ところに・～ところへ relate to timing, but from different angles. ついでに expresses performing a second action while it's convenient, e.g. '買い物のついでに郵便局に寄った' ('I stopped at the post office while shopping'). ～ところに・～ところへ focuses on when something happens, often unexpectedly, e.g. '出かけようとしたところに電話がかかってきた' ('Just as I was about to leave, the phone rang').

Both connect actions to a larger context, but with different emphasis. ついでに expresses performing a second action while it's convenient, e.g. '買い物のついでに郵便局に寄った' ('I stopped at the post office while shopping'). にあたり・にあたって is more formal and used for actions on a certain occasion, e.g. '開会にあたり、ご挨拶申し上げます' ('On the occasion of the opening, I extend my greetings').

ついでに and がけに both relate to doing something while engaged in another action. ついでに expresses a second action done while convenient, e.g. '買い物のついでに郵便局に寄った' ('I stopped at the post office while shopping'). がけに is used with movement verbs and emphasizes doing something during a transition, e.g. '帰りがけに電話した' ('I called on my way home').

Both can be translated as 'while', but with different implications. ついでに means 'while you're at it' and adds a convenient extra action, e.g. 買い物のついでに銀行に寄った ('I stopped by the bank while shopping.') うちに emphasizes doing something before a condition changes, e.g. 熱いうちに食べてね ('Eat it while it's hot.')

Both ついでに and 際に refer to additional actions tied to another event, but differ in tone. ついでに is an everyday expression for when something is done while convenient, e.g. '買い物のついでに郵便局に寄った' ('I stopped at the post office while shopping'). 際に is more formal and emphasizes a particular occasion, e.g. '来日の際にご連絡ください' ('Please contact us when you come to Japan').

Both can mean 'while,' but the nuance differs. ついでに implies doing something extra during another task, e.g. 買い物のついでに郵便局に寄った ('While shopping, I stopped at the post office.'). ているあいだに just marks that something happened during another action, as in 買い物しているあいだに電話がかかってきた ('I got a call while shopping.').

ながら attaches to verb stems to express performing one action simultaneously with another, e.g. 音楽を聴きながら走る 'listen to music while running'. In contrast, ついでに follows various words or phrases to convey the idea of taking advantage of the opportunity to do another thing at the same time, e.g. 買い物に行くついでに走ってきた ('On the occasion of going shopping, I went for a run.').

ているところだ and ついでに both relate to ongoing actions but differ greatly in usage. ているところだ describes an action in progress at that moment, with the nuance that the speaker is currently occupied (e.g. 昼ご飯を食べているところだ = 'I'm in the middle of eating lunch'). In contrast, ついでに describes performing an 'extra' action on the same occasion as another (e.g. 散歩のついでに昼ご飯を買ってきた = 'While on a walk, I bought lunch').