First, are TIP and AS even about the same subject? The question may seem odd; surely they are both about international politics (or world politics)? Bull writes at the outset that "this book is an inquiry into the nature of order in world politics" (AS, p.xi); Waltz writes that his aim is "to construct a theory of international politics that remedies the defects of present theories" (TIP, p.1). So Bull's focus, at first glance, might seem narrower: he says he is concerned "not with the whole of world politics but with one element in it: order" (AS, p.xi). However, the notion of 'order' he uses is general enough to undergird a discussion that, in its own way, is as sweeping as Waltz's. Both books are big-picture "grand theory," albeit very different examples of the genre. Waltz is self-consciously constructing a parsimonious theory that he claims meets "philosophy-of-science standards" (TIP, p.1), whereas Bull is not interested in constructing a theory of that kind (or, arguably, of any kind). Waltz's theory is a 'systems theory' in that it gives special importance to (one particular definition of) the structure of the international system as distinct from the 'units'; Bull's approach, while focusing on system-wide institutions that the 'units' themselves have created and through which they regulate their relations, is not a 'systems theory,' at least not in the Waltzian sense.
The past and continuing preoccupation of many IR theorists with the notions of 'system' and 'structure' has sparked a reaction by some (e.g., R. Jackson, The Global Covenant, p.31: "There is no international 'system' or 'structure' that exists and functions outside human decision, responsibility and control"), but the allure of 'structure' -- now often reformulated as 'networks' -- remains quite strong. (I won't address networks here, nor will I discuss the "practice turn" in IR theory, which perhaps has some connections to the English School.)
Waltz and "structure"
Structural realism is structural because it holds that the most important thing to know about international politics is the distribution of power across (or among) states, and this distribution is considered a "structural" rather than a "unit-level" property. Thus, according to this way of thinking, the fact that the U.S. is the most militarily powerful country in the world is not considered a fact about, or a property of, the U.S.; rather, it is viewed as an aspect of the current system's structure. As Waltz put it: "How units stand in relation to one another, the way they are arranged or positioned, is not a property of the units. The arrangement of units is a property of the system." (TIP, p.80) "The distribution of capabilities is not a unit attribute, but rather a system-wide concept." (p.98)
Like many of his realist predecessors, Waltz stresses that the "ordering principle" of "anarchy," i.e., an absence of central authority or world government, means that states (the 'units') ultimately can look only to themselves to protect against (real or perceived) threats and to ensure their survival. The result, in his view, is a strong tendency for balances of power to form over and over, as states find it necessary to prevent the most powerful state in the system from becoming so powerful as to threaten their respective existences as independent entities.
Thus two main "expectations" of Waltz's theory are that "balances of power recurrently form, and states tend to emulate the successful policies of others." (p.124) A problem with the first of these expectations or predictions is that it doesn't seem to match up very well with fairly large swaths of history. Waltz tries, to some extent, to anticipate this objection by stressing the difficulty of testing theories, especially those which yield general rather than specific expectations, and by noting that "[b]ecause only a loosely defined and inconstant condition of balance is predicted [by the theory], it is difficult to say that any given distribution of power falsifies the theory." (p.124) He cautions in the opening chapter that "the rigor and complication of tests must be geared to the precision or to the generality of the expectations inferred from the theory." (p.16) He never says explicitly 'don't subject this theory to overly rigorous tests because its expectations are general not precise,' but he comes very close to saying that.
Another main point readers usually take away from Theory has to do with "the stability of a bipolar world," to quote the title of Waltz's 1964 article on that subject. Partly in the interest of keeping this post to a reasonable length, I won't address that aspect of the book here.
Note (1): The degree to which the entire 'realist tradition' is 'structural' in its emphases is a debatable question. For one perspective on the issue, see J. Parent & J. Baron, "Elder Abuse: How the Moderns Mistreat Classical Realism" (International Studies Review, June 2011).
Note (2): Waltz's definition of 'structure' is obviously not the only one possible. Contrast, for example, the view that "international structure consists fundamentally in shared knowledge...." (A. Wendt, Social Theory of International Politics [STIP], p.31)
Bull and the "element of society"
Unlike Waltz, Bull doesn't have to worry, at least not explicitly, about theory construction and testing because he doesn't see himself as doing science (see the so-called 'second great debate'). So whereas Waltz begins with a chapter about what a theory is, Bull doesn't need one. A separate point is that Bull rejects the idea of "value-free" social inquiry (see AS, p.xv), but he doesn't elaborate much on this, at least not in meta-theoretical terms, in the book.
As is well known, Bull distinguishes between an international system, in which states interact enough that "the behaviour of each [is] a necessary element in the calculations of the other" (p.10), and an international society, in which states, "recognising certain common interests and perhaps some common values,...regard themselves as bound by certain rules in their dealings with one another...." (p.13) As is also well known, he aligns himself with what he labels (aptly or not) "the Grotian tradition," which emphasizes the "element of co-operation and regulated intercourse among states." (p.41) It coexists, in different degrees at different times, with 'Hobbesian' and 'Kantian' elements (respectively, "state of war" and "transnational solidarity and conflict"). (pp.41,51)
The heart of The Anarchical Society is Part 2, where five institutions -- the balance of power, international law, diplomacy, war, and the special role of the great powers -- are assessed in terms of their contributions to 'international order'. This is preceded by a chapter on "Order versus Justice." Quoting a passage from that chapter (p.97) will give a taste of Bull's style and also show how normative considerations are woven into his analysis:
...not only is order in world politics valuable, there is also a sense in which it is prior to other goals, such as that of justice. It is does not follow from this, however, that order is to be preferred to justice in any given case. In fact ideas of both order and justice enter into the value systems, the justificatory or rhetorical stock-in-trade of all actors in world politics. The advocate of revolutionary justice looks forward to a time when a new order will consolidate the gains of the revolution. The proponent of order takes up his position partly because the existing order is, from his point of view, morally satisfactory, or not so unsatisfactory as to warrant its disturbance. The question of order versus justice will always be considered by the parties concerned in relation to the merits of a particular case.For Waltz, the international system is a case of "order without an orderer and of organizational effects where formal organization is lacking." (TIP, p.89) To elucidate these characteristics Waltz looks to "microeconomic theory" (ibid.), in which actors' normative beliefs or commitments are basically irrelevant. For Bull, by contrast, actors' values influence how they behave, which in turn influences system-level outcomes.
It's sometimes overlooked that Bull in AS sees international society as only one element of international politics. If you're a grad student writing a comprehensive exam, the statement that the English School "treat[s] the international system as a society governed by shared norms" (to quote Wendt, STIP, p.31) will get you through. However, in a brief section called "The Limitations of International Society" Bull writes that the element of international society "is always in competition with the elements of a state of war and of transnational solidarity or conflict" and thus "it is always erroneous to interpret international events as if international society were the sole or the dominant element." (p.51) The word "always" here seems too strong; why foreclose the possibility that there may be periods in which the element of international society is "dominant"? Whether that is the case today is a question best left for another occasion.