What's the difference between a "conflict of interest" and the "appearance of a conflict of interest"?
If I'm a waiter, and I'm responsible for the distribution of tips from the tip jar at the end of the night, that's a conflict of interest.
If I'm a waiter, and I'm responsible for pushing the button that distributes tips from the computerised register, with no reasonable opportunity to interfere or influence that automatic distribution, you might argue that I still shouldn't do it since this would be an appearance of a conflict of interest.
In other words, people like to avoid suspicion due to the the appearance of a conflict of interest no matter how unreasonable that suspicion might be.
A conflict of interest is such that one's interests conflict with whatever is at issue. For example, he must choose an employee from among a set of candidates, and one of the candidates is his relative, about whom he feels strongly enough that that feeling affects his judgement.
An appearance of a conflict of interest is such that one's interests appear to conflict with whatever is at issue. For example, he must choose an employee from among a set of candidates, and one of he candidates is his relative, about whom we don't know that he feels anything affecting his judgement (so we don't know there's a conflict of interest — but there's definitely the appearance of one).
It is often — as in the latter case — impossible even for a person possible affected by a conflict of interest to know whether there is such a conflict.
A conflict of interest is when you are in a position to take advantage of your position. So, if your wife is also one of the vendors you are considering to buy from, that is a conflict of interest.
An apparent conflict of interest is avoiding what could look like a conflict of interest, even if it makes no sense at all. So if you work for certain US government sub-contractors, and travel on business, you are not allowed to stay over the weekend and pay for that yourself, even though the subsequent air fare would be cheaper for the company if you stayed over Saturday. It LOOKS like you're having fun at taxpayer expense, and they want to avoid that apparent conflict of interest, even if it costs taxpayers more.