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In general, a qubit is mathematically represented as a quantum state of the form $\lvert \psi\rangle = \alpha \lvert 0\rangle + \beta \lvert 1\rangle$, using the basis $\{ \lvert 0\rangle, \lvert 1\rangle \}$. It seems to me that a qubit is just a term used in quantum computing and information to denote a quantum state (i.e. a vector) of a system.
Is there any fundamental difference between a qubit and a quantum state? What's more to a qubit than the quantum state it represents?
There are a few things to distinguish here, which are often conflated by experts because we're using these terms quickly and informally to convey intuitions rather than in the way that would be most transparent to novices.
A "qubit" can refer to a small system, which has a quantum mechanical state.
The states of a quantum mechanical system form a vector space. Most of these states can only be distinguished from each other only imperfectly, in that there is a chance of mistaking one state for the other, no matter how cleverly you try to distinguish them. One may then ask the question, of a set of states, whether they are all perfectly distinguishable from one another.
A "qubit" is an example of a quantum mechanical system, for which the largest number of perfectly distinguishable states is two. (There are many different sets of perfectly distinguishable states; but each such set contains only two elements.) These may be
the polarisation of a photon ($\lvert \mathrm H \rangle$ versus $\lvert \mathrm V \rangle$, or $\lvert \circlearrowleft \rangle$ versus $\lvert \circlearrowright \rangle$);
or the spin of an electron ($\lvert \uparrow \rangle$ versus $\lvert \downarrow \rangle$, or $\lvert \rightarrow \rangle$ versus $\lvert \leftarrow \rangle$);
or two energy levels $\lvert E_1 \rangle$ and $\lvert E_2 \rangle$ of an electron in an ion, which may occupy many different energy levels but which is being controlled in such a way that the electron stays within the subspace defined by these energy levels when it isn't being acted on.
Common to these systems is that one can describe their states in terms of two states, which we might label as $\lvert 0 \rangle$ and $\lvert 1 \rangle$, and consider the other states of the system (which are vectors in the vector space spanned by $\lvert 0 \rangle$ and $\lvert 1 \rangle$) using linear combinations taking the form $\alpha \lvert 0 \rangle + \beta \lvert 1 \rangle$, where $\lvert \alpha \rvert^2 + \lvert \beta \rvert^2 = 1$.
A "qubit" can also refer to the quantum mechanical state of a physical system of the sort we've described above. That is, we may call some state of the form $\alpha \lvert 0 \rangle + \beta \lvert 1 \rangle$ "a qubit". In this case we are not considering what physical system is storing that state; we are interested only in the form of the state.
"A qubit" can also refer to an amount of information which is equivalent to a state such as $\alpha \lvert 0 \rangle + \beta \lvert 1 \rangle$. For instance, if we know two states $\lvert \psi_0 \rangle$ and $\lvert \psi_1 \rangle$ of some complicated quantum system, and we have some physical system whose state $\lvert \Psi \rangle$ is in some superposition $\alpha \lvert \psi_0 \rangle + \beta \lvert \psi_1 \rangle$, then it doesn't matter how complicated the system is or whether either of the states $\lvert \psi_j \rangle$ have any entanglement: the amount of information expressed by the possible values of $\lvert \Psi \rangle$ is one qubit, because with a clever enough noiseless procedure, you could reversibly encode that complicated quantum state into the state of a (physical system) qubit. Similarly, you can have a very large quantum system which encodes $n$ qubits of information, if you could reversibly encode the state of that complicated system as the state of $n$ qubits.
This may seem confusing, but it's no different from what we do all the time with classical computation.
If in a C-like language I write int x = 5; you probably understand that x is an integer (an integer variable that is), which stores an integer 5 (an integer value).
If I then write x = 7; I don't mean that x is an integer which is equal to both 5 and 7, but rather that x is a container of sorts and that what we are doing is changing what it contains.
And so forth — these ways in which we use the term 'qubit' are just the same as how we use the term 'bit', only it so happens that we use the term for quantum states instead of for values, and for small physical systems rather than variables or registers. (Or rather: the quantum states are the values in quantum computation, and the small physical systems are the variables / registers.)
Quantum systems are identified with Hilbert spaces $\mathcal{H}$ (with certain properties).
Quantum states are identified as trace-1 positive semidefinite operators on this space $\mathcal{H}$. In your case, pure states are also rays in $\mathcal{H}$ (That's when you could write them as kets). As such, when you talk about quantum states you should specify the Hilbert space.
Now, qubit is the name given to the specific Hilbert space $\mathcal{H} = \mathbb{C}^2$, which is a two-dimensional space.
