Just to clear up some notational confusion (there doesn't seem to be any completely standard notation):
In Bernstein's notes:
The "easy" pullback (i.e. the one that coincides with the pullback of the underlying $\mathcal O$-module) is denoted $Lf^\Delta$.
$f^! = Lf^\Delta [dim X - dim Y]$ (right adjoint to $f_!$)
$f^\ast = \mathbb D f^! \mathbb D$ (left adjoint to $f_\ast$)
Note that $f^\ast$ and $f^!$ agree with the corresponding functors for constructable sheaves (not for the underlying $\mathcal O$-modules).
However... In Hotta, Takeuchi, Tanisaki
The easy pullback is denoted $Lf^\ast$ to agree with the $\mathcal O$-module functor.
Berstein's $f^!$ is now called $f^\dagger$.
Bernstein's $f^\ast$ is now called $f^\star$
Ok, now in David Ben-Zvi's answer above (and in many other places)
The easy pullback is $f^\dagger$, and the rest agrees with Berstein's notation.
In my opinion, the most important notational feature to be preserved is that $(f^\ast , f_\ast)$ and $(f_! , f^!)$ form adjoint pairs.
To answer your question
When $f$ is smooth, $f^\ast = \mathbb D f^! \mathbb D = f^! [2(dim Y - dim X)]$, and the easy inverse image functor is self dual (and preserves the t-structure).
One way to think about these different pullbacks is that the easy inverse image preserves the structure sheaf $\mathcal O_Y$. This corresponds to the constant sheaf shifted in perverse degree under the RH correspondence. On the other hand $f^\ast$ preserves the "constant sheaf" whereas $f^!$ preserves the dualizing sheaf (for a smooth complex variety, these correspond to the D-modules $\mathcal O[-n]$ and $\mathcal O[n]$).