What's the difference, if any, between fylgja and hamingja?
I'm still a little messed up in the head from a cold but I'm gonna take a crack at this, just be advised that there's really no way to summarize this in a blog post no matter my cognitive state so I'll just try to get you started by focusing on the problems of differentiation in the border between the two concepts. I'm going to draw heavily from Zuzana Stankovitsova's MA thesis on fylgjur, and if you want to understand the depiction of fylgjur in the sagas I highly recommend you read the whole thing.
This is kind of a constantly ongoing discussion. The problem isn't that there isn't a discernible difference, it's that each one is internally inconsistent, and within those spans of inconsistency, there's some overlap. So you get instances of fylgjur and hamingjur that seem to have hardly anything to do with each other, but then also separately instances of things called fylgja that you'd have expected to be called hamingja, etc. This honestly is very common in Norse literature generally. We expect consistent, distinctive, taxonomic categories but you can't do a DNA test on a norn or an isotope analysis on a dís's fossilized skeleton. But with fylgja the fuzziness might be a little above average even for Norse concepts. It also probably changed over time, within the span of time documented in the sagas.
The crux of the confusion is the fact that while the fylgja most often appears as an animal, sometimes there are scenes in the sagas where a figure in the form of a woman (usually in a dream or vision or some other exceptional, non-normal context) is called fylgja, which resembles what we would expect to be called hamingja. Most scholars have treated animal-fylgjur and woman-fylgjur completely separately.
I'm at least partially inclined to agree with that approach. I think the word fylgja (which just means 'follow') gets thrown around a little much, and that not everything referred to with that word was ever meant to be seen as the same thing. In modern Icelandic the word fylgja can even refer to just a regular ghost that haunts a person (rather than a place, so it follows them from place to place; and an ættarfylgja haunts successive generations of a single family). When fylgjur appear in sagas in the form of human women, they sort of bleed into other categories like hamingjur or dísir and on the extreme end maybe even things the word valkyrja might apply to. If the word fylgja can be applied to any figure that follows a person, then a hamingja is a type of fylgja, just one that is different from the animal-fylgja.
To demonstrate, here are two very similar scenes from two different sagas:
In Hallfreðar saga, Hallfreðr suddenly gets deathly ill. His fylgjukona 'fylgja-woman' appears to him, big and wearing armor, able to walk on the sea as if on land (characteristic of valkyries in Völsunga-related mythology), and Hallfreðr declares that he is formally severing ties with her. Then the woman asks Hallfreðr's son Þorvaldr whether he would like to accept her. He says no, so she asks Hallfreðr's other son, who consents, and she becomes his fylgjukona.
In Víga-Glúms saga, Glúmr dreams of a huge woman, so big that her shoulders take up the entire breadth of valleys, touching mountains on either side. In the dream, he went outside to greet her and bid her welcome into his home. When he woke up, he interpreted this as meaning that his maternal grandfather had died and that his hamingja had come to Glúmr.
I don't personally find it clear whether these are supposed to be the same "type" of being or not. I think the easiest way for someone to try to simplify this would be to say that the author of Hallfreðar saga simply chose confusing wording, and could have said hamingja, or that fylgjukona is not the same as an animal-fylgja. I'm not gonna make that call, as I have no problem with the idea that the evidence really just is confusing and contradictory and I prefer not to give into the impulse to systematize.
Most often, when a figure is described as a fylgja, it's a sort of animal double. They're often deployed in the story to foreshadow death -- an animal representing a main character will be seen in a dream being attacked by other animals representing their enemies. A recurring motif is that people will be suddenly overcome with fatigue in the middle of the day and be unable to help falling asleep, in order to have these extraordinary dreams. People with special abilities, or people in highly unusual situations, may be able to see them without being asleep (I think this only happens once in the literature we have). In rare instances, these visions serve as warnings that enable the relevant people to actually avoid the fate they were headed toward (making it very different from other kinds of knowing the future in Norse literature, which in almost every other case is unavoidable).
The animal fylgja (or other things like it) has a place in later Scandinavian folklore but with all of the change, development, and speculation around it it's not possible to systematize or summarize it; this is a whole continuum of belief and not a single cohesive thing I can summarize. If you're interested in that a good start in English would be Scandinavian Folk Belief and Legend, edited by Reimund Kvideland and Henning K. Sehmsdorf. It's also very easy to conflate the fylgja with other kinds of animal affinity, like Kveldúlfr becoming a wolf at night or seiðmenn projecting their awareness outside of their bodies in the form of animals, but most scholars have considered this separate and I agree with them.
Despite the fact that hamingja is a little less all over the place, I find it even harder to describe. Usually the word just means 'luck' or 'happiness' in a general sense and doesn't refer to a discrete being or personality, yet the etymology suggests that, at least when the word started being used, it was applied to a specific sort of non-physical being. The word is believed to come from ham-gengja, something which "goes" (ganga) and is of hamr (roughly 'shape' or 'form' but a complicated discussion on its own).
Presumably this has made you have more questions than you started with but hopefully I have at least moved the locus of confusion somewhat.
For some further reading (for convenience, including the ones I already mentioned):
Kvideland, Reimund and Henning K. Sehmsdorf, eds. Scandinavian Folk Belief and Legend
Murphy, Luke John "Herjans dísir: Valkyrjur, Supernatural Feminities, and Elite Warrior Culture in the Late Pre-Christian Iron Age"
Sommer, Bettina Sejbjerg. "The Norse Concept of Luck"
Stankovitsova, Zuzana, "'Eru þetta mannafylgjur?' A Re-Examination of fylgjur in Old Norse Literature" (see also her bibliography)
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