What are the similarities and differences between Von Braun's proposed mission to Mars and SpaceX's?

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No one knows much about SpaceX’s plans yet.

Only two components are known at all: the MCT, or Mars Colonial Transporter, which is an interplanetary craft that can deliver 100 tons of payload (including passengers) to the Martian surface; and the BFR, or Big Fucking Rocket, which can put the MCT on a Mars intercept and is reusable.

The BFR will use a large array of Raptor engines, which are fueled by methane and use a sophisticated full-flow staged combustion system. If built as expected, it will be by far the largest rocket ever constructed, beating the Saturn V by a factor of 3 or more. The engines are not individually very large, implying that they’ll need a large number of them. The lower and upper stages will be fully reusable, and their fuel and engine choices are centered around having a high cycle count.

Little else is known. They are supposed to announce some details later this year.

I don’t know much about Von Braun’s plans, other than that there were several iterations and that they called for hundreds of launches and many thousands of tons of payload (the later iterations were scaled down somewhat).

Orbital mechanics being what they are, there cannot be much difference in the general mission timeline–9ish months to get there; aerobrake at Mars; descend to the surface (leaving the return craft in orbit); stay there for a bit over a year; finally, ascend and return to Earth (also taking 9ish months). The devil is in the details, and Von Braun worked out some of them in his books, but there is vastly more to work out for any real mission.