instance
The noun INSTANCE has 2 senses:
1. an occurrence of something
2. an item of information that is representative of a type
The verb INSTANCE has 1 sense:
1. clarify by giving an "example" of
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example
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If you ask someone for "an instance", you are asking them to tell you about something that actually happened -- something real.
If you ask for an "an example", it might be something real or something theoretical which serves as an illustration of something.
Technically:
I would hear for example to refer to a particular type of occurrence where many types exist.
And I would hear for instance to refer to a particular occurance of the same type, where many occurances exist.
For example:
The auditors look for many types of fraudulent activity, for example, reporting revenue from fictional sales. Implying there are other types of fraudulent activities.
Mr X is being investigated for wash sale rules violations, for instance the trade with ABC Corp where he bought the same shares in his personal account from DEF Corp. Implying there are other violations of the same type, ie wash sale violations, and this is just one instance/occurance.
In principle, they are not synonymous. However, in practice they are used interchangeably.