Are there any downsides to using a deadcat, even indoors when there is no wind, compared to a foam windscreen?

There may be a small HF difference but I tend to keep mine on all the time as if you bang the mic off the ceiling or floor it protects it better than a foam windgag.

The furry deadcat, which is typically installed over the foam, may get dirty easier, costs much more than foam if you have to buy/replace it, adds a bit of weight, makes the mic look larger, and is one more piece of gear to fuss with. The acoustic effects, in the absence of wind, should be minimal, but you would have to judge whether or not they are a factor in you application. I only use one under windy or potentially windy conditions.

I agree with what has been said but I've filmed indoors without protection and then in the middle of a theater performance the central air comes on.

OK. I ran a quick test as follows:

Mic - A Rode NTG2 with their supplied foam and deadcat.
Phantom power and capture via TSCAM US-399
Captured into Adobe Audition 3.0 at 44.1 sample rate
Signal used for comparison was 10 seconds of white noise generated by Audition played over some KLH 2-way small speakers.
Mic in shock mount on stand about 24" from the speaker, about 12" of axis.

Average RMS levels
Mic only taken: as 0 dB
mic with foam: --2.7 dB relative to mic only
mic with foam and deadcat: -6.8 dB relative to mic only (-4.1 relative to mic and foam)

However, it is not that simple.
Did a frequency analysis of the three options and note that most of the difference is in energy at frequencies above around 6 kHz. Below that frequency there is not much difference (allowing for the nature of white noise). I suppose a sweep of a pure tone might have been better but I didn't have time to set it up.

My take on this is that the dead cat (and foam for that matter) will hurt high frequency components but would not be an issue for lower frequencies and voice (e.g., AM radio bandwidth).

I have attached two screen grabs showing the frequency variation for about 9 seconds of the test.

With respect to central air condition; the cat would help if the mic is in a strong draft or air stream, but would not likely help the low frequency rumble common through the room.

Make that a TASCAM US-366.

NTG2 is a pretty average mic but having had an NTG1 leaving the softie on will probably improve the sound quality as they tend to sound quite fizzy with the usual midrange electret suckout.