In order to obtain an explicit description of the diffeomorphism, one can work as follows.
Take $B=\mathbb{C}^{n-1}$, with coordinates $t_1, \ldots, t_{n-1}$, and consider the complex space $\mathcal{X}$ obtained glueing $\mathbb{P}^1 \times \mathbb{C} \times B$ with $\mathbb{P}^1 \times \mathbb{C} \times B$ by the identification of
$(y_0, y_1, z, t_1, \ldots, t_{n-1})$ with $(y_0', y_1', z',t_1, \ldots, t_{n-1})$
if
$z'=z^{-1}, \quad y_1'=y_1z^{-n}, \quad y_0'=y_0+y_1 \Sum_{i=1}^{n-1}t_iz^{-i}$.
Let us denote by $\pi \colon \mathcal{X} \to B$ the family obtained in this way.
Let now $T_k \subset B$ be the determinantal locus given by
[ T_k= \left( \begin{array}{ccc} t_1 & \ldots t_{k+1} \\ t_2 & \ldots t_{k+2} \\ \cdot \cdot \cdot \\ \cdot \cdot \cdot \\ \cdot \cdot \cdot \\ t_{n-k-1} \ldots t_{n-1} \end{array} \right) ]
Then, if $t \in T_k - T_{k-1}$ we have $X_t \cong \mathbb{F}_{n-2k}$.
By using Ehresmann theorem, one concludes that $\mathbb{F}_n$ and $F_{n-2k}$ are diffeomorphic.