What's the difference between a "racist" and a "racially sensitive" remark?

"Racially-insensitive" is the bigot-sensitive version of "racist".

Edit: let me add a bit of less-flippant elaboration:

Usually when someone makes a "racially sensitive" remark, he's being "racially insensitive". A racially sensitive remark is one that touches upon sensitive points, so being insensitive (towards race or any other touchy subject) means not being aware of the sensitivity of the issue.

The reason I call this phrasing "bigot-sensitive" is that it avoids calling someone "racist" by saying he's merely "insensitive to the sensitivity of the issue".

In this case, a racist remark is one where insult is meant by saying the words. A racially sensitive remark is one that uses words that could be insulting or abusive, but depending on the context.

Note the square brackets indicating omission and/or insertion. The BBC are tip-toeing around the words Judge Howard Riddle used in his summation by removing them and replacing them with a watered-down summary.

The judge actually wrote in his judgment (PDF linked from the article):

There is also no dispute that John Terry directed the words “black cunt” in the direction of Anton Ferdinand.

It's not the BBC's place to use the word racist as they would then be making a judgment of guilt about John Terry. In fact, the judge also said:

... the issue for this court to decide is not whether Mr Terry is a racist, in the broadest sense of the word... The issue between the defendant and the Crown is whether Mr Terry uttered the words “fucking black cunt” by way of insult. If he did then the offence is made out, regardless of what may have motivated him.

Additionally, the court reached a not guilt verdict. The judge wrote of Terry:

It is therefore possible that what he said was not intended as an insult, but rather as a challenge to what he believed had been said to him.

In those circumstances, there being a doubt, the only verdict the court can record is one of not guilty.

Instead, like most news sources, the BBC don't want to publish the words directly in their article (but have no problem publishing the judgment document which repeats them many times), and have referred to them obliquely.

The word "sensitive" in the phrase "racially sensitive" means "touchy."

sensitive = calling for tact, care, or caution in treatment

The difference between "a racist remark" and "racially sensitive words" is that "racist" can mean a conscious, deliberate, and unrepentant act of insulting.

The expression "racially sensitive," on the other hand, is open to the interpretation and possibility that “what was said was not intended as an insult.”

When I hear "Racist remark", I imagine a remark that places a person's ethnicity as the cause of their failing, or in some way contributing to their inadequacy, or part of a verbal attack because of their race.

"Racially sensitive remark" makes me think it was a word that, in itself, is not a direct insult towards a person. For example, the word "nigger" is considered racially sensitive because of the history around the word, not necessarily because it confers any meaning (aside from that the person is black).