I was wondering about cohomological and k-theoretical Euler class, or both versions of characteristic class in general. I mean, one knows that characteristic classes can measure, how twisted such a bundle is and for particular bundles we concider appropriate characteristic classes. But they are based on cohomology theoery usually. So my question is, what is measured by Euler classes, that is founded as the k-theoretical one? There are examples that both versions gives different results and one cannot force one result (cohomological for example) to the second one (k-theoretical). So maybe these two versions of Euler class measure two different 'factors' or rather one of them is more, let's say, precise and therefore results are different (which mean, more precise and accurate)?
Can anyone give me any hint or present another approach, or explanation how both versions can be understood?