Why can’t a qbit be both entangled and in a pure state?
First, let's look at the density matrix for A in the two cases. When A is in the pure state $|+\rangle$, we have
\begin{align}
\rho_+ = |+\rangle\langle+| = \frac{1}{2}\left(|0\rangle + |1\rangle\right)\left(\langle0| + \langle1|\right)
\end{align}
As a matrix, we'd write this
\begin{align}
\rho_{+} = \frac{1}{2}
\begin{pmatrix}
1 &amp; 1\\1&amp; 1
\end{pmatrix}.
\end{align}
When A is entangled with B, we can obtain the (reduced) density matrix for A by tracing over all B states:
\begin{align}
\rho_A &amp;= \frac{1}{2}\text{Tr}_B\left[\left(|0\rangle\otimes|0\rangle + |1\rangle\otimes|1\rangle\right)\left(\langle0|\otimes\langle0| + \langle1|\otimes\langle1|\right)\right]\\
&amp;= \frac{1}{2}\left(|0\rangle\langle0| + |1\rangle\langle1|\right),
\end{align}
or, as a matrix,
\begin{align}
\rho_{A} = \frac{1}{2}
\begin{pmatrix}
1 &amp; 0\\0&amp; 1
\end{pmatrix}.
\end{align}
Ok, you say, the density matrices are different. Who cares? Well, we don't really care about states, because we can't measure states directly. But we can measure observables, and we can find the expectation value for those measurements using our states. Specifically, we can find the expectation value of $\mathcal{O}$ by taking $\langle\mathcal{O}\rangle = \text{Tr}\,(\rho\mathcal{O})$. Let's look at the example of the $X$ operator. We have
\begin{align}
\langle X\rangle_+ &amp;= \frac{1}{2}\text{Tr}\,(\rho_+ X)\\ &amp;= \frac{1}{2}\text{Tr}\,\left[ \begin{pmatrix}1&amp;1\\1&amp;1\end{pmatrix}\begin{pmatrix}0&amp;1\\1&amp;0\end{pmatrix}\right]\\
&amp;=\frac{1}{2}\text{Tr}\,\begin{pmatrix}1&amp;1\\1&amp;1\end{pmatrix} = 1.
\end{align}
In the state $\rho_A$, however, we have
\begin{align}
\langle X\rangle_A &amp;= \frac{1}{2}\text{Tr}\,(\rho_A X)\\ &amp;= \frac{1}{2}\text{Tr}\,\left[ \begin{pmatrix}1&amp;0\\0&amp;1\end{pmatrix}\begin{pmatrix}0&amp;1\\1&amp;0\end{pmatrix}\right]\\
&amp;=\frac{1}{2}\text{Tr}\,\begin{pmatrix}0&amp;1\\1&amp;0\end{pmatrix} = 0.
\end{align}
We obtain different experimental predictions for the outcome of measurements in these two states, and so we can distinguish between them.

Edit: I didn't really answer your question beyond showing how states in the example you gave lead to different predictions. I would say the more broad lesson to draw is that when system A is entangled with system B, measurements on B can change the outcome of measurements on A. This leads to an additional, classical uncertainty in our measurements that is not present when A is in a pure, unentangled state.