A little googling would answer that already. The type of work done, the fee segment you work in, the kind of clients you get exposure to and the brand value are just different.
First, the work in Big 4 is different. You specialize in one area e.g. ops, m&a, IT, accountancy etc.. In Strategy consulting you'd hit a broad spectrum of projects from operations to tenders and strategy.
Second, Big 4 works in a different fee segment. This means that the kind of clients you get exposure to is different than in strategy consulting.
Also in Big 4 you will work with more specialized clients e.g. the IT security department of the client rather than on Cx0 level projects.
Third, the reputation of MBB is that of ultra elites. Much like an Ivy League school, the brand names will open doors regardless of your on the job performance.