This is a great question; I hope I can write a half-decent answer. Some of these points have already been made by others, but I wanted to go into more detail.
Atmospheric escape
I actually don't think that the atmospheric composition will change significantly, because the solar wind isn't the primary driver of atmospheric escape. Rather, photons - particularly, ultraviolet photons - are the main energizers of molecules. You can certainly transfer energy from solar wind particles via a sort of analog to sputtering, but it's not the dominant mechanism. If you're not convinced, then consider the "luminosity", so to speak, of the solar wind - the change in kinetic energy imparted by the Sun onto the wind. It's
$$L_{\text{wind}}=\frac{1}{2}\dot{M}v_{\infty}^2$$
with $\dot{M}$ the mass-loss rate and $v_{\infty}$ the speed. Assuming $\dot{M}\sim10^{-14}M_{\odot}\text{ yr}^{-1}$ and $v_{\infty}\approx500\text{ km s}^{-1}$, I get that
$$\frac{L_{\text{wind}}}{L_{\odot}}\approx2\times10^{-7}$$
meaning that most of the energy the Sun impacts on molecules in the atmosphere is transferred via photons, not the solar wind. Therefore, removing the solar wind wouldn't affect thermal escape, the dominant category of mechanisms of atmospheric escape.
Interplanetary medium vs. interstellar medium
It's hard to compare the interplanetary medium to the interstellar medium (ISM) because the ISM is a very complicated thing (which, of course, astronomers are more than happy to ignore from time to time!). There are three stable phases (McKee & Ostriker 1977) and a myriad of structures, including molecular clouds, galactic cirrus, H II regions, and more. They vary drastically in temperature (from $10\text{ K}$ to $\sim10^6\text{ K}$) and density (from $\sim10^{-2}\text{ cm}^{-3}$ to $\sim10^3\text{ cm}^{-3}$).
The Solar System is (likely) embedded in a structure called the Local Interstellar Cloud (LIC), a structure inside the larger Local Bubble, so let's say that your system is, too. The LIC has a temperature of $7000\text{ K}$ and a number density of $n\sim0.1\text{ cm}^{-3}$. This makes it roughly an order of magnitude cooler and a factor of 50 less dense than the interplanetary medium. In other words (as others have said), if the interplanetary medium was replaced with the ISM immediately outside the Solar System, nothing would change for the planets - and that assumes that the Sun wouldn't just heat it up again, which seems rather likely.
The key point here is that there really wouldn't be any issues from the (quite small) influx of particles from the ISM - not by a long shot. We'd probably see some differences in elemental abundances, but there's still going to be mainly hydrogen and dust around.
The auroras
I can't add anything here; you're quite right that we would no longer see auroras. Interactions between the solar wind and a planet's magnetosphere are crucial for their production, and without one of those . . . well, you don't get much of a light show.
Comet tails
In the comments, DWKraus noted that comets would be affected. Comets have two main tails (as well as a fainter tail of escaping sodium), one composed of ions pushed away by magnetic fields and the charged particles of the solar wind, and the other composed of dust blown away by solar radiation pressure. Without the solar wind, the tail of ions would be gone, but the dust tail would remain.
