Which are the biggest methodological differences between the archaeological and geological approaches to stratigraphy?
As I see it there are no clear differences. There are, however, processes that are not compatible with a superposition principle. In geology, it is possible for layers to be folded and result in inverse age relationships. Packs of sediments can also be pushed over other layers during, for example mountain building, so that sequences are out of order or even repeated, almost like shuffling a deck of cards. It is also possible for, for example, magma to penetrate into rocks and generate much younger layers within an older rock mass. A depth, mass is added to the crust by melting and solidification while cooling such that rock units of different age can be found parallel (in a stratigraphic context) to each other. Hence the superposition principle only applies in sedimentary rocks and unconsolidated sediments that has not undergone and significant tectonic activity. But, I can also imagine that archaeological material can (although not extensively) be subject to disruptions in certain environments where active tectonics are at play or for example where glaciers act to push material forward and generate repeat sequences. 

To expand on the second part of the post. Geological stratigraphy is established from physical exposures and by measuring structural shapes such as the strike and dip of strata (a branch of geology called Structural geology. Very little is gained from created exposures such as open day mines, mine shafts of other dug pits. It is also possible (as in archaeology) to use geophysical methods to infer layer by ground penetrating radar, seismics etc. It should be mentioned here that methods and possibilities vary between bedrock and unconsolidated sediments, where I would venture to say that archaeology and studies of the unconsolidated sediments are very similar using test pits and coring with the addition of understanding transport mechanisms to understand how and why certain sediment layers exist where they are.

Visualisation pof geological data is usually done in sections or logs of boireholes with attempts to connect between observed sections. In bedrock where there is not only vertical variability but also significant horizontal, such as in igneous rocks, obtaining a precise image is very difficult