ON and German have the same four cases standard to Germanic case languages, nominate, accusative, dative, and genitive. They differ considerably in the usage of the latter two, though.
German tends to be very reserved in the usage of dative and especially genitive; genitive in German is used only to mark possession. ON has, however, a much more evenly spread case usage, with dative and genitive seeing much use; genitive can be observed to follow prepositions and even verbs. Dative also follows many verbs whose meanings would seem direct enough to call for a simple accusative. Some examples of all this:; ; hrinda + dat = to push someone (note how it doesn't take accusative, which would seem logical)
In German, umlaut is comparatively easy and convenient to deal with. ON features a much more extensive umlaut; in fact, umlaut pervades the whole morphological system of the language. The ON orthography also does not account as well for the umlaut as German does. ON umlauts are of two types, i-umlaut and u-umlaut, identified by which sound originates the umlaut. The u-umlaut is simple, only effecting the change a > ö, but can appear unpredictably, because in many cases the u that originated the umlaut is no longer present. Examples of u-umlaut:
In fact, the main difficulties with this is how inadequately the orthography accommodates the umlaut, compared to German orthography.
ON umlaut appears just about anywhere. There are many correspondences with German here, such as plurals, derived verbs, adjective comparatives, all having i-umlaut. But generally the occurrences are too many for me to attempt to relate here; the umlaut will have to be tackled as it appears.