To transform from a standard (2D) Cartesian coordinate system to polar coordinate system, we have the relations
$$ r = \sqrt{x^2+y^2},$$ $$ \theta = \arctan{\frac{y}{x}},$$
for the vector components. We also have the relations
$$ \hat{x}=\cos{\theta}\,\hat{\theta}-\sin{\theta}\,\hat{r},$$ $$ \hat{y}=\sin{\theta}\,\hat{\theta}+\cos{\theta}\,\hat{r},$$
for the unit vectors. If we have an arbitrary vector $$\vec{v}=x\hat{x}+y\hat{y}$$ in Cartesian coordinates, what is the difference between transforming the vector using the first set of transformations, i.e.
$$ \vec{v} = \sqrt{x^2+y^2}\,\hat{r}+\arctan{\frac{y}{x}}\,\hat{\theta}$$
and transforming the unit vectors using the second set of transformations, i.e.
$$\begin{align} \vec{v}& =x\left(\cos\theta\,\hat{\theta}-\sin\theta\,\hat{r}\right)+y\left(\sin\theta\,\hat{\theta}+\cos\theta\,\hat{\theta}\right) \\ & = \left(y\cos\theta-x\sin\theta\right)\hat{r}+\left(y\sin\theta+x\cos\theta\right)\hat{\theta}\,, \end{align}$$
and why is one of these incorrect?
The issue has to do with the fact that the first set of transformations are simply transforming the components of the vectors. They are not vectors themselves, but they show how the components $(x,y)$ transform into $(r,\theta)$. The second set of transformations shows how to transform a basis of the vector space into another basis.