The first possible plastic component is the fuselage. From a pilot operation point of view, you can guess that flying metallic, fabric or plastic fuselage won't change anything. This is correct.
There are only two differences to be noted: planes with plastic fuselage are available in white, and possibly light gray. This is to avoid excessive sun heating that darker color could produce. There is not a long history on composite evolution over long term in aviation, but other industries use it for years now, so this restriction is probably based on strong reasons. However, this is not really restrictive, except for pilots liking pink or greenish planes...
Plastic fuselage also need different defrosting / deicing methods. Scratching frost or using certain de-icing fluids could severly damage the surface.
I must also report that one of the DA40 in my new homebase had to return recently to Vienna, because it has a so-called "gelcoat" problems. In fact, some bubbles formed under / in the paint. This was serious enough to have Diamond calling it back to the factory. On the positive side, Diamond made a replacement plane available. This is still a pending issue, so I will publish when everything will be fixed.