With a question such as this one, it is useful to specify just what you mean by Marxism, because there is Marxism as a school of socialism (predominantly associated with communism), but there is also Marxism as a school of historical and sociological analysis, which has the aim of explaining and understanding historical, political, and social trends through the lens of class struggle and economic factors.
The two are linked, especially given that Marx himself understood history as progressing towards an eventual classless society, but even so, Marxism as an analytic framework and Marxism as a political system are not, strictly speaking, the same thing and should not be conflated with one another. For this answer, as others have already spoken about the purely political dynamic, I will be speaking toward Marxism's advantages and disadvantages as an analytic framework.
In many respects, one of the greatest advantages of Marxism, especially within the context of nineteenth- and even early-twentieth-century intellectual circles, was the degree to which it broadened and expanded our understanding of historical causation, along with human political and social interactions. For example, the histories produced in this time period (and earlier) were primarily narrative based and often primarily concerned with the dominant political figures and events of any age. Marxist analysis thus actually served to deepen our understanding of historical causation, introducing new dynamics such as class and economic factors. Under the Marxist historians, you would see increased focus on financial or tax policies, for example, or on the experience of agrarian peasants or the urban poor.
However, at the same time, one of its great strengths was also its greatest weakness as an analytic model. Marxism is a very powerful mechanism for exploring historical and sociological causation, but it also tends to be a totalizing narrative: for Marxists, everything tends to be driven by class struggle and economic factors. This has the effect, however, of sidelining other competing influences: the role of contingency, for example, with the degree to which individuals or even pure chance can serve as hinges on which history turns, not to mention entire approaches to historical analysis such as intellectual and political history, to give two examples. In the process, Marxist analysis risks becoming a form of analytic dogma, trapped in its own presuppositions.