Personally, I think there's a huge difference.  I think of the very popular and current term, "pro-active."

How can there not be a difference between doing something and not doing something?

Obama seems to be very good at ignoring things, but who knows how much active work goes on behind the scenes to create the impression that something is not even on his radar screen??

As they say, "I digress!"

Is there a difference between de-toxification and not toxifying?

To "not emphasize" something is to ignore it.

To "de-emphasize" something is to actively take steps to counteract (probably) existing factors which, if left unchecked, would bring that thing to the forefront.  This flies in the face of the ancient wisdom, "Just ignore it and it will go away."  In other words, you tried that, and it didn't work.  So now you have to become pro-active.

Political "damage control" is a science.  You have to be very clever to insure that the actions you take don't simply call more attention to the thing you're trying to soft-pedal. (Methinks the lady protests too much!)  Are we just lying about it?  Or possibly denying something, as a ploy to call wanted attention to it?

Sorry this is so rambling.  I'm not really awake yet.

Nothing is ignored by the "operatives" behind the scenes of a political campaign.  No stone is left unturned in their efforts to create the impressions they wish to leave in the minds of the voters.

When your theoretical speaker says, "Obama de-emphasized in his campaign that race was an issue," does he mean that in his speeches, Obama said things to suggest that race was not an issue in his campaign?   or that he ordered his campaign workers to refuse to take advantage of situations where race could work in his favor?  You'll have to ask your theoretical speaker what was in his mind.

Hi,
My friend, a native speaker, insisted that most native speakers would think that 99% of the time a person who uses "de-emphasize" in his sentence means "not to emphasize". I agree that sometimes people speak in this way, but I certainly wouldn't say 99% of the time. Perhaps the percentage depends on whether you spend most of your time with educated or uneducated people .

Let's review by considering 3 very simple examples.

Tom went to a job interview.

1. In the interview, Tom emphasized his experience. This means Tom kept saying that he had a lot of great experience.

2. In the interview, Tom did not emphasize his experience. He did not say much about it.

3. In the interview, the interviewer emphasized Tom's experience. He said that he thought Tom had a lot of great experience. However, Tom said that his experience was not important. Tom de-emphasized his experience. In other words, in this example the experience was emphasized by someone else and then de-emphasized by Tom.

Best wishes, Clive