The English language verbs Intrude and Interfere have similar meanings, but they are different, as I shall explain, using British English examples.
Intrude means to go into a place where you should not be. This could be as simple as causing someone inconvenience by being there, right up to it being illegal to do so. You can intrude into a physical location, but also a virtual place, such as a meeting or conversation.
"The cleaner intruded into the important meeting."
"We can't just go in without asking, that would be intruding."
"I am having a private conversation, please don't intrude."
"The intruder, caught on CCTV, turned out to be the burglar we were looking for."
Intruder (noun) is the one who intrudes.
"No one noticed the intruder for several minutes."
"When I saw the back door was open, I knew there had been an intruder. "
"The intruder left a mess in the hall."
Interfere means to take part in or interrupt an action without authority or necessity, and thereby be a nuisance to someone or be unhelpful. You can't interfere with something over which you have control and rights. For example you can't interfere with your own child's education. You can interfere with someone else's. Those with the best intentions are often those who interfere the most.
"Please don't interfere with the control panel."
"My brother is not going to interfere with your plans."
"My husband's mother is always telling us how to dress our baby, she interferes in everything!"
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