Most adcoms look at your highest score. This is the number they report for marketing purposes and rankings.

The management schools consider your best GMAT score during admission. Therefore, your score of 680 will be valid.

Regarding the sufficiency of the score, so anything above 670 is largely considered okay. In your case, if you have a promising professional profile and stellar essays to match up to your GMAT score of 680, you may be standing quite fine.

As a standard, the worth of a GMAT score is ascertained by the batch's GMAT average. Many MBA schools declare the middle 80% GMAT score range. Therefore, if your score falls within or above this range, your chances of selection are high.

However, in the worst case, if your score falls below this range, do apply, but your chances will be quite uncertain here.

Draft engaging and effective essays. Market yourself well in them. Convince the adcom. This works the best!

I'm going to disagree with the two poster but it's probably because of semantics. The schools will SEE & CONSIDER all of your scores. The degree to which they do that varies from school to school. Trust me on this one...

When they REPORT your GMAT score they will use your highest score.

To that end, I wouldn't worry about your 650 the 2nd time.