Denotatively (ie, in terms of strictly the definitions):
Therian = someone who identifies as an animal (sometimes you'll see this defined as earthen animals only, but this is historically not necessarily part of the label - if it's animalistic, it qualifies)
Otherkin = someone who identifies as nonhuman (be that animal, plant, sapient nonhuman people, object, concept, etc.)
Connotatively (ie, in terms of associations and cultural nuances), there are some cultural distinctions as well. The therian and otherkin communities actually arose completely independently, and only later started interacting so closely that in many ways they've nearly merged.
The otherkin community originates from a few separate groups of elves, namely the Silver Elves and the Elf Queen's Daughters, both of whom originate somewhere in the 1970s (exact dates often difficult to pin down) and which then led to the Elfinkind Digest, a mailing list originally intended for elves but which quickly expanded to other nonhuman beings, within which the term otherkind (which would rapidly become otherkin, as far as I can tell due to typographical errors that accidentally got preserved in the lexicon) was coined to avoid the inconvenience of having to write elfin-/dragon-/orc-/hobbit-/etc-kind constantly within the Digest. The influence of these groups and communities has meant the otherkin community has always had heavy influence from New Age and neo-pagan philosophies, and has always leaned toward spiritual explanations - though certainly never exclusively.
The therian community, on the other hand, originates from a messaging board called alt.horror.werewolves, often abbreviated to AHWW within the therian community these days, in the early 1990s. While this messaging board was originally a fansite for werewolves, at a certain point people started turning up who felt themselves to be (were)wolves under the skin, and gradually that kind of discussion took over much of the board. (Not just wolves, either - some of the earliest weres I remember seeing in logs include a bat and, of course, the now-famous werePontiac.) If you've ever seen animal-people, particularly older ones, calling themselves weres instead of/in addition to therians, this is why - that was the original word we used for the concept. Culturally speaking, the therian community has always been very focused on the concept of shifts because of these origins - it is, if I'm not mistaken, where the otherkin community adopted that terminology from. The therian community also tends to be more focused on the animality of being non-human - which is why these days that's usually used as the primary distinction between otherkinity and therianthropy.
Ultimately, there's a lot of overlap, but there are also a lot of reasons why someone might want to use one word or the other even if both could theoretically apply to them denotatively.
