the most relevant dimensions based on my experience of serving clients for McKinsey and BCG:
Expertise. Sometimes this is being questioned, but MBB consulting firms are indeed specialized in a large array of industries and business functions. When adding up the experience of their consultants from junior business analysts to senior partners and knowledge experts, they can bring thousands of years of experience in each particular sector to their clients. Additionally, top firms spend millions in industry-leading research each year. Lastly, they are able to tap into the best external knowledge via expert platforms.
Pool knowledge across functions and across levels. Consultants are not part of the client's culture, politics, or organization culture. In the first month, as the firm builds a fact base, consultants usually interview people across functions. In large companies, cross-functional problem-solving rarely happens. Just getting different functions in a room typically unlocks creative problem solving. Similarly, consultants interview, watch, and tag along with people down the organization's structure, often starting with customers and moving through sales and line roles. CEOs and the exec team of large companies rarely do this (exception being their largest customers). There are tremendous insights to be had by doing this.
Problem solving & methodological toolkit. Consultants are the most experienced problem solvers in the business world. They have established processes, tools as well as access to data and experts to deal with any question a company, government or NGO could possibly imagine. From my own experience, it is just amazing how the drive of a small motivated and experienced team of hard working problem solvers can identify and solve business problems, that the client they are working for is unable to solve within a much longer timespan and a much bigger workforce.
Outside perspective. Consultants can bring an objective outside perspective into the client organisation. Since they serve mostly very large corporations or governments, the clients are prone to suffer from management inertia, groupthink, and daily routines. Bringing fresh and objective thinking from the outside can go a long way in such environments to crack crusted structures and old paradigms.
Driving change. As an external force, they are often called to evaluate a company’s issues, recommend solutions and lastly, help overcome or circumvent internal politics to drive change in the organization. Consultants are not bound by the client’s internal political landscape or other issues. Therefore, their substantiated advice can often be used as a powerful force of change. Additionally, especially McKinsey have made huge strides to increasingly support the implementation of their recommendations and have established a rich set of "new delivery models".
Capability building. Besides the analytical work a typical consulting engagement often includes, consultants can spend a significant amount of their time to train employees of their client, either in 1-on-1 coachings or full-fledged workshops. They transfer skills, build up knowledge to help bring their solutions into the organization and facilitate the implementation. Consultants usually train people from senior client leadership to sales staff and shop floor workers.
Political leverage. CEOs that want or need to make an unpopular decision often bring in a consulting firm to help. This provides ammunition to recommend an unpopular or risky decision to the board (expansion into a new business line or geography, or shutting down a plant). The CEO can also distance herself from an unpopular decision by blaming the consultants. Finally, if things go wrong, consultants are a handy scapegoat who won't complain too loud.