ABS (should) allow for the best stopping traction you can get (in a perfect world) on the road surface regardless of road conditions.
That said, you see some caveats in the mix there. I'll try to explain.
I say should allow for the best. All ABS systems are not created equal. Some work better than others (though I'm not going to break those out). ABS design should allow for the wheels to break traction with the road surface for only a smallest amount of time (hundredths or tenths of a second). As a tire loses traction, it stops rotating. The wheel speed sensor picks this up and tells the ABS pump to cycle on that wheel which allows the wheel to start spinning again. Then the cycle starts again. The computer is checking all four wheels at the same time (for most ABS systems ... some only have one sensor for the rear brakes, which relies on the spinning of the differential to tell it when it's moving or not. This is not a big deal as over 60% of the braking is done on the front end and the front is where the turning occurs, so traction during stopping is way more important there.) There is a situation on ice, if all four wheels were to stop spinning at the same moment (complete lockup), the ABS system will not react, because all four wheels are rotating at the same rate (ie: not spinning at all). This would make for the car to slide uncontrollably, unless you can get the tires to start rolling again, which means releasing then reapplying the brakes, which means great stopping distance. Ice is a special case, though. The regular recommendation is to not drive on ice, but hopefully that is only common sense.
Would my stopping distances increase?
For the most part, yes, your stopping distance will increase over a similarly equipped vehicle with ABS present. The reason is because the ABS pump can cycle much faster than you can pump the brakes.
Does it make a lot of difference in good conditions (e.g dry, smooth roads)?
ABS does nothing for stopping distance up to the point of where the tire break traction with the road surface. When that happens, the ABS kicks in and does its job, which greatly reduces your stopping distance. When the road surface is good, the ABS has less of a chance to activate, even during hard braking. That is because the grip between your tires and the road surface is much greater. Your tires will usually start to lose grip at the point I'd call emergency braking ... where you really need to stop fast. This is when you can expect the ABS to kick in and this is the point at which you need it the most.
In terms of stopping in an emergency situation (i.e. maximum pressure on the brake pedal), what are the other effects of not having ABS?
One of the greatest dangers with your wheels locking up, which ABS pretty much solves, is that when the front tires lock up, you lose the ability to steer your vehicle. During emergency stopping, without ABS, your front tires would have the tendency to lock up. When this happens, you can try and steer all you want, but your momentum will carry you in the same direction. The natural tendency of the driver at this point is to press on the brakes harder, trying to get your car to stop faster, but the car has now lost control and you are just along for the ride. If you are an experienced enough driver, you can overcome this tendency and pump the brakes. As you can see, this will definitely make for longer stopping times. Even on dry roads, ABS can make a difference.
Would it make any difference in icy conditions?
It does to an extent, but you still have to be careful when driving on icy roads, as I stated above.