What are the main specific differences between F1 and Indy cars. Such as weight limits, engine restrictions, fuel, etc.
There are many differences between F1 cars and Indy/Champ cars, too many to list here but the basic difference is that F1 cars are more technologically advanced than Indy cars, as you would expect as F1 is the pinnacle of motorsport.
Current F1 cars have hybrid engine systems including ERS (Electronic Recovery Systems) that harvest energy such as braking energy and allow the driver to deploy this as an extra boost to performance, typically F1 cars generate around 750 HP plus an extra 100 HP or so when the ERS is being deployed.
The rules for 2015 mean all F1 teams have V6 turbo charged engines, with a capacity of 1.6 litres, fuel used is "normal" petrol/gasoline.
Indy cars use Ethanol as their fuel and their engines are 2.2 litre V6 twin-turbo, up to 700 HP power output (Plus push to pass, extra ~100 HP). So engine power is fairly similar for both types, the F1 car having a small advantage.
F1 cars are very light, they can have a minimum weight of 600 kgs under the 2015 rules and so with driver and fuel included they are around the 750 kg mark at the start of the race (including up to 100 kg of fuel). Indy cars are around 25% heavier at ~750 Kg plus driver and fuel. This is one of the main reasons F1 cars are quicker around a circuit compared to Indy cars as they can accelerate more quickly (better power to weight ratio) and also stop more quickly. They are also more nimble with quicker cornering speeds due to better aerodynamics and shorter wheelbase. Indy cars are designed (largely) for oval circuits which means they have a longer wheelbase to give more stability but this makes then slightly less agile.
Acceleration for an F1 car (typical) - I cannot find similar data for Indy Cars but they will be slower due to the extra weight etc.
0-100 Kmh / 62 MPH 1.7 s 0-200 Kmh / 124 MPH 3.8 s 0-300 Kmh / 186 MPH 8.6 s
Top speeds for both cars are fairly similar, around the 240 MPH mark. F1 cars are geared to whatever circuit they are racing at though, so a lot of tracks they will be topping out at significantly lower speeds, ie 170 MPH is typical at Monaco (no long straights) whilst around Monza it is over the 220 MPH mark. Indy cars generally race at Ovals and faster circuits so they will achieve their top speed more often.
F1 has carbon fibre brakes which perform a lot better than the steel brakes used by Indy cars.
In 2006 both F1 and Champ Cars raced around the Circuit Gilles Villenuve, Montreal, Canada. The F1 cars were 5-7 seconds a lap quicker than the Indy cars, pole was set by Fernando Alonso with a 1m 14.942 lap and the Indy Car pole was 1m 20.005s set by Sebastien Bourdais. The fastest laps under race conditions were slightly further apart with the F1 cars around 7 seconds a lap quicker.
F1 teams will run on a budget of around £200M per year (for the top teams such as Ferrari) compared to a budget around £15M for a top Indy Car team such as Newman/Haas.
Where possible I am referring to current ie 2015 rules but some examples are from different years, in which case I have given the relevant year.
Most of this has come from Wikipedia but I have used other sources as well. I will add to this answer wheere possible as I want to get the accelleration figures for Indy cars but that's not very forthcoming!