Germanic is just a term describing languages related to modern German. Norse describes a subset of Germanic languages, specifically those related to modern Danish, Swedish, Norwegian, and Icelandic. So all Norse speakers speak a Germanic language, but not all Germanic speakers speak a Norse language. Using these linguistic terms to describe cultures, however, can get a bit hairy.

Scholars used to assume that people and language were directly related, so if you spoke a Germanic language, they thought it meant that you were ethnically German. This doesn't entirely make sense. For example, even though millions of Indians speak English, it wouldn't make much sense to think of them as ethnically English or as adherents of a uniquely English faith (e.g. Anglicanism). Some undoubtedly are, but calling them all English would be pretty confusing.

But many historians still use "Germanic" to describe not just Germanic languages but also the peoples that spoke those languages, as well as the things they produced (e.g. pottery, metalwork, and architecture) and sometimes even the lands where they lived. Others prefer to split fine hairs. Even if a people speaks a Germanic language, these scholars argue, we shouldn't call them Germanic because it suggests that they have a common "Germanic" identity or maybe even descended from common "Germanic" ancestors.

This conflation of language, identity, and descent can become a huge problem in the hands of nationalist groups. The term "Germanic" has been especially fraught due to its association with the Nazis. Nazi Germany used the presence of Germanic speakers in the Sudentenland to justify annexing territory, they used the archaeological discovery of "Germanic" pots to justify invading their neighbors, and they used the concept of an ancestral "Germanic" homeland to justify ethnic cleansing within their borders. Obviously there were other issues at stake, but at least some historians are very cautious about the term (and others like it) because of nationalist legacies.

These hesitations aside, the terms "Germanic" and "Norse" do frequently appear in scholarship, and it would be absurd to ignore or try to discard them. So in rapid review:

Germanic-speaking cultures. During the Roman period, this generally referred to peoples living in modern-day Germany, whereas the British Islands and Gaul (France) were populated by people who spoke Celtic languages. During the early medieval period (400-1100), people also began to speak Germanic languages in the British Isles (Anglo-Saxons), France (Franks), Spain (Visigoths), Italy (Ostrogoths and Lombards), and North Africa (Vandals). Many people in these areas still continued to speak languages related to Latin, and these "Romance" languages eventually came to predominate in France, Spain, and Italy. Arabic eventually superseded Germanic and Romance languages in North Africa. By 1100, Germanic-speaking cultures were limited to the areas of modern Germany, the British Isles, the North Atlantic, and Scandinavia.

Norse-speaking cultures. These seem fairly limited to Scandinavia throughout the early middle ages (also referred to as the Iron Age especially in Scandinavian scholarship). Between 800 and 1000, however, Norse-speaking colonists had spread across the North Atlantic, and Norse-speaking elites ruled over much of the North Sea basin and the Russian river ways. This resulted in a lot of linguistic borrowing in northern England, but less (that I'm aware of) in Slavic- or Romance-speaking areas. After 1100, political fortunes changed, and Norse languages generally constricted to the areas of modern Scandinavia and Iceland.

I think there is a lot to learn about the rise and fall of languages and linguistic groups, and using words like "Norse" and "Germanic" help us define our studies and communicate with other scholars. But we also need to remain attentive to the cultural variety they can subsume.

For example, we shouldn't think that the Visigoths and the Viking Rus' participated in the same religion, just because they're both Germanic: the Visigoths practiced a type of Christianity known as Arianism, whereas the Rus' practiced an extreme form of paganism (or so their Muslim ethnographer Ibn Fadlan would have us think). Speaking a Germanic language doesn't mean practicing a "Germanic" religion (even if we were to assume that a "pure" Germanic religion did at one time exist).

Nor should we assume that Melkorka in Iceland and Harald Hardrada shared a common origin just because they both ended up speaking Old Norse: Melkorka had been raised as an Irish princess, whereas Harald got fifteen years of his education under the Byzantine administration as a member of the Varangian Guard. Speaking a Norse language doesn't mean a common Norse origin, nor should it even suggest that Norse-speaking cultures provided the most influential formative experiences for a given individual.

So they're not quite the same, the distinction can be useful, but we also need to be careful in how we use the terms.

I'm going to answer from the perspective of my field, which is not history but linguistics.

"Norse" is a subset of "Germanic." Germanic culture and language spread from a nexus in northern Germany starting well before Tacitus, and by the late 2nd century CE it was already heterogeneous enough that the branch of the language family spoken in Scandinavia was recognizably distinct. I presume, though somebody specializing in very early Germanic cultures can probably clarify, that the Scandinavian Germanic peoples were also culturally distinct by this point. (The organization of the Germanic languages and their place within the wider Indo-European family is well enough documented to be considered minimally interesting; much of what you need to know about it is found in any introductory textbook in historical linguistics, of which my favorite is Larry Trask.)

But that doesn't make them any less Germanic. Early Saxons, Franks, and Goths were all culturally and linguistically distinct from one another as well. (There are examples of things like Swabian hairstyles in which men grew their hair long and tied it in a large loose knot which are not found in any other region; that one is mentioned in Tacitus, Germania 1:38.) The speakers of Proto-Norse, the language that would later develop into Old Norse and thence into the modern Scandinavian languages, were merely one of several kinds of Germanic people; any generalization that applies across the Germanic-speaking world necessarily includes them, but there are facts about Norse peoples that are specific to the Norse, just as there are facts about the Goths that are uniquely Gothic.

The other part of the answer is that when we talk about "Germanic" culture we usually mean early in its spread, as that's when the Romans were encountering the various Germanic groups. When we talk about the "Norse" culture, we usually mean speakers of Old Norse (the last common ancestor of the modern Scandinavian languages); that was spoken during the Viking age, which started a couple centuries after the (western) Roman Empire had fallen. Cultures change a lot in that timeframe, so even though they were entirely Germanic, and it's a mistake to conflate the Viking Age Norse with Classical-era speakers of Proto-Norse as described in Roman sources.