K-theory and usual (cohomoogical) Euler class Can someone explain the difference (and hence the sense of studying) between the usual and K-theoretical Euler class? Most famous and for now the only difference is showed during computations of tensor product of two line bundles.

Namely $e^K(L_1\otimes L_2)\neq e^H(L_1\otimes L_2)$, where $e^K$ and $e^H$ are K-theory and cohomology Euler classes.

Any intuition or maybe an article?
The Euler classes for ordinary cohomology and K-theory are examples of characteristic classes. Roughly speaking, characteristic classes take vector bundles over a space $X$ and give you classes in $E^*(X)$, where $E^*$ is some suitable generalized cohomology theory in the sense of Eilenberg-Steenrod.

Some people study these generally because they give us a homotopy theoretic handle on geometry. Others study them because they give us a geometric handle on homotopy theory. There's a lot to be gained from the interplay between these two areas.

Specializing now to the homotopy theory side, we study characteristic classes because it turns out that, quite often, a large amount of data about the cohomology theory is packed in the characteristic classes. This leads (eventually) to the chromatic picture of homotopy theory, which is beautiful and mysterious.
The Euler classes for ordinary cohomology and K-theory are examples of characteristic classes. Roughly speaking, characteristic classes take vector bundles over a space $X$ and give you classes in $E^*(X)$, where $E^*$ is some suitable generalized cohomology theory in the sense of Eilenberg-Steenrod.

Some people study these generally because they give us a homotopy theoretic handle on geometry. Others study them because they give us a geometric handle on homotopy theory. There's a lot to be gained from the interplay between these two areas.

Specializing now to the homotopy theory side, we study characteristic classes because it turns out that, quite often, a large amount of data about the cohomology theory is packed in the characteristic classes. This leads (eventually) to the chromatic picture of homotopy theory, which is beautiful and mysterious.