I feel like this is really a broad question, because 1) it sort of depends on the particulars, and 2) interpretations can vary. This is especially the case with Hegel.

I think the clearest point of agreement between Hegel and Marx is in Hegel's Master-Slave dialectic.

But the basic run-down that you'd get from some kind of intro book is that Hegel's logic proceeds in accord with the following structure: Thesis-Antithesis-synthesis.

The thesis in some sense "contains" the anti-thesis. But because of a one-sided approach that is not immediately apparent. The thesis is a "moment". There is somehow a move to the anti-thesis, which negates the thesis. However, this too is only a moment, and because of the same one-sided approach ignore the fact that the anti-thesis also "contains" the thesis. They negate one another yet rely upon one another for definition. Only when this is realized does a double-negation happen -- which is termed "sublation" -- whereupon you arrive at the synthesis between two opposing ideas.

Now, if you're full-blown Hegel, this sublation to a synthesis is also its own thesis, and the dialectic continues.

But that's the sort of quick and dirty run-down you'd find in attempting to convey some point of general agreement on Hegelian dialectics.

Right.

I'd add: The primary difference between Hegel and Marx being that the former was an idealist, while Marx spearheaded dialectic materialism. So for Hegel it pertains primarily to the realm of ideas, while Marx saw the dialectic in real-world relations.

For example, how capitalism (thesis) can bring about socialism (synthesis) because maximum efficiency in factories most often involves bringing groups of people together, i.e. fostering social cohesion (antithesis).

/deleted for pedantry

It might be helpful to think of the dialectic in Marxist terms as a driving or motor for change. The implications behind this are several, as a materialist, mentioned above Marx sees the engine of change as social factors, particularly the relationship of different social groups to the means of production. For him class is not an attitude, way of thinking, construct it is practically determined ( a source of much criticism of Marxist interpretations of events such as the French Revolution..( I could say more) ) Marx is also a determinist, along with several other mid 19th thinkers in other fields such as Darwin, Macaulay, he sees some governing meta narrative, for Darwin evolution, Macaulay the rise of liberal parliamentary democracy. This of course reduces the role of the individual who is not so much a decisive actor as something carried by the grand tide ( Iron Laws as Marx would say) of history. In dialectical terms collision is the motor for change, the idea ( Hegel) generates its opposite and out of this clash emerges the new idea( thesis) For Marx this collision is class conflict and the new idea is the new set of material relations eg feudalism gives way to capitalism and the dominance of those controlling the means of production, banks, farms, mills etc. However for Marx change is not endless, the proletarian revolution involves the overthrow of the capitalist system and the destruction of its supportive state apparatus, once all is held in common there is no us and them, thesis and antithesis , socialism becomes the end state ( condition). hope this helps.

Marx's dialectics according to David Harvey:


'One of the most important things to glean from a careful study of Volume I is how Marx's method works. I personally think this is just as important as the propositions he derives about how capitalism works, because once you have learned the method and become both practiced in its execution and confident in its power, then you can use it to understand almost anything. This method derives, of course, from dialectics, which is, as he points out in the preface already cited, a method of inquiry "that had not previously been applied to economic subjects" (104). He further discusses this dialectical method in the postface to the second edition. While his ideas derive from Hegel, Marx's "dialectical method is, in its foundations, not only different from the Hegelian, but exactly opposite to it" (102). Hence derives the notorious claim that Marx inverted Hegel's dialectics and stood it right side up, on its feet.' (Harvey 2010: 11)

In what immediately follows the above, Harvey makes it clear that Marx's dialectics is to understand "processes of motion, change and transformation."

'There are ways in which, we'll find, this is not exactly true. Marx revolutionalized the dialectical method; he didn't simply invert it. "I criticized the mystificatory side of the Hegelian dialectic nearly thirty years ago," he says, referring to his critique of Hegel's Philosophy of Right. Plainly, that critique was a foundational moment in which Marx redefined his relationship to the Hegelian dialectic. He objects to the way in which the mystified form of the dialectic as purveyed by Hegel became the fashion in Germany in the 1830s and 1840s, and he set out to reform it so that it could take account of "every historically developed form as being in a fluid state, in motion." Marx had, therefore, to reconfigure dialectics s that it could grasp the "transient aspect" of a society as well. Dialectics has to, in short, be able to understand and represent processes of motion, change and transformation. Such a dialectical method "does not let itself be impressed by anything, being in its very essence critical and revolutionary" (102-3), precisely because it goes to the heart of what social transformations, both actual and potential, are about.' (Harvey 2010: 11)


Harvey goes on to explain how Marx's dialectics is different from Hegel's and the version of Marx we tend to believe in.

'What Marx is talking about here is his intention to reinvent the dialectical method to take account of the unfolding and dynamic relations between elements within a capitalist system. He intends to do so in such a way as to capture fluidity and motion because he is, as we will see, incredibly impressed with the mutability and dynamics of capitalism. This goes against the reputation that invariably precedes Marx, depicting him as some sort of fixed and immovable structuralist thinker. Capital, however, reveals a Marx who is always talking about movement and the motion - the processes - of, for example, the circulation of capital. So reading Marx on his own terms requires that you grapple with what it is he means by "dialectics." '(Harvey 2010: 11-12)

One example how Marx's dialectics works is seen when he explains the dual aspect of value of the commodity: use-value and exchange value.

'Let us reflect a moment on the structure of this argument. We begin with the singular concept of the commodity and establish its dual character: it has a use-value and an exchange-value. Exchange-values are a representation of something. What is it a representation of? A representation of value, says Marx. And value is socially necessary labor-time. But value doesn't mean anything unless it connects back to use-value. Use-value is socially necessary to value.'
(Harvey 2010: 22)

The dialectics is not about causality, at least uni-directional causality. It is about codependent relations that requires its constituents at the same time.

'How has Marx's dialectical method been working here? Would you say that exchange value, or use-value cause ...? This analysis is not causal. It is about relations, dialectical relations. Can you talk about value without talking about use-value? No. In other words, you can't talk about any of these concepts without taking about the others. The concepts are codependent on one another, relations within a totality of some sort.' (Harvey 2010: 33)