Can someone give me a brief rundown and comparison between the ideas of Clausewitz and Jomini regarding military strategy? Wiki's not making it clear enough for me.
Sure.
Jomini came first, and was initially much more popular than Clausewitz (whose book was only widely read outside Prussia after the 1860s, when Helmuth von Moltke referred to it as the foundation of German military success). Although he is known for popularizing some
concepts, like maneuvering on interior lines, Jomini's most enduring contribution to military theory is his effort to fit warfare into a more logical system: one that consisted of problems to be solved, and appropriate solutions to those problems dictated by the laws of war.
Clausewitz's book is much more all-encompassing.
Vom Kriege has sections on things that Jomini never really touched upon, like the interaction between the conduct of war and the political leadership of the state, the experience of war by soldiers and to a greater extent commanders, and the impact of the so-called fog of war, sometimes referred to as "[Clausewitzian] friction". Clausewitz was explicitly nondogmatic, in sharp contrast to Jomini, and argued that context was supreme in each individual war, that they were all products of their times. He was the first to tentatively explore the relationship between tactics and strategy, although he never formulated any concept of operations himself. And he was a philosopher and teacher, who employed a sort of dialectic to try to push readers towards their own informed conclusions rather than devising dicta outright in many cases. This has led to wildly varying opinions about his work and what was most relevant in it. For pre-WWI German officers, it was his analysis of the relationship between tactics and strategy; for American officers in the period of the Revolution in Military Affairs, it was his discussion of the relationship between politics and strategy.
There's no real need to set the writers in opposition to each other, as formulating alternative and mutually exclusive conceptions of war. Jomini was first and foremost a military theorist; Clausewitz was theorist, philosopher, historian, and teacher all wrapped up in one. While certain of Clausewitz's themes can be read in opposition to Jomini's - chiefly, Clausewitz's insistence on not being dogmatic and on the importance of fog, friction, and moral forces - others synthesize reasonably well.