I've taken the RER from CDG to Paris during non rush hour and it was great.

However, after a long flight I can get claustorphobic and even my carry on seems a burden. Just how crowded does the RER get at this hour? For example, in comparision to the Picadilly line, Heathrow to central zone?

ttt

In the morning, the RER lines are crowded going into Paris; in the evening, it's the other way around. They can get crowded enough that RATP employees stand around to push people into the cars, Tokyo-style.

Taking crowded public transport is not a good idea if you want to stay healthy for your stay. There's no better way to catch a cold than to stand in public transport at rush hour.

I used to take that RER line to and from work and it does get very crowded indeed. With luggage it wouldn't be much fun and I wouldn't recommend it.

If you want to take public transport rather than a taxi, then you could get the Roissybus which is a non-stop bus service into Paris (it drops passengers on Rue Scribe in the 9th, by the Opera house). It won't be nearly as full and is more comfortable when travelling with luggage, though it will take longer than the RER.
You can also get the Air France shuttle buses, which go to Gare de Lyon/Montparnasse or Porte Maillot/Etoile.

Thank you very much, AnthonyGA and hanl. That was just what I was looking for: how crowded is the RER rush hour crowded! My appreciation for your replies.

If you take the RER that goes directly to Gare du Nord without stops then it will be virtually empty all the way to that stop. Every other RER is direct from the airport.