I have what is probably a terrible way of approaching linear vs. holistic, but it's all I've got:
Linear: "Given what just happened, where do I look next for the answer?"
Holistic: "Given all that's happened, what's really going on here?"
Very often in detective stories, the expression of either problem solving method is illustrated through a "you're looking at this wrong" moment. Imagine two detectives are staring at the body of a murdered man whose gold watch was stolen:
Dumb detective: "Looks like somebody wanted a gold watch real bad."
Genius detective: "Don't be foolish. You're missing all the other elements. Look how vicious the stabbing was . . . how the body was placed face up even though the wounds were to the back . . . there's much more to this than a gold watch."
I'd argue (perhaps incorrectly?) that the above makes the genius detective a holistic problem solver. You could just as easily find a story in which . . .
Dumb detective: "What a vicious killing . . . the placement of the body is bizarre . . . I can't even begin to imagine where we'd start looking for the killer. Should we look for "
Genius detective: "Don't be distracted by nonsense and extraneous details. Just follow the money: who wanted that watch?"
I'm happy to be corrected on all this, but it always seems to me that the holistic problem solver tends to see the complexity and seek to find how everything fits together, whereas the linear problem solver sees the straight line and seek the most logical answer.