This is the most obvious difference between adult koi and goldfish. If you see a colorful fish in a pond that is larger than 15 inches, it is a koi. Fishbase.org lists 12.5 in (32 cm) as the maximum size for goldfish while it lists 43 in (110 cm) as the maximum size for koi. Younger specimens in ponds can be more difficult to tell apart, though, so this isn't always reliable.
Goldfish are pretty limited in their patterns and colors whereas koi have a much wider variety. Goldfish tend to come in solid colors of orange, white, black/blue, or red. Yellow can be seen but is very uncommon. White and black combinations and red and black combinations are also possible but rare compared to the common red and white combination. Koi come in a much greater variety of common combinations. The base colors are red, white, black, yellow, orange, and blue. The chart below illustrates some common koi colorations and the names of the patterns.
|Koi color varieties (credit in picture)|
The barbels (sometimes called whiskers)
This is the most obvious difference between koi and goldfish. Koi will have these barbels and goldfish will not. If you can see these on a pond fish, that fish is a koi. However, these aren't always easily observable.
|Barbels on koi (Stan Shebs @ Wikipedia)|
Both koi and goldfish can have elongated tails, but only goldfish will have the signature double-tail. This is where the caudal fin (tail fin) is doubled. From above, this will often give the tail a three-pronged or a four-pronged appearance. This characteristic is only found in goldfish.
|Typical split caudal as seen on wakin goldfish (fishsempai.com)|
Other less obvious physical characteristics
This illustration below points out more of the morphological differences between koi and goldfish. Often times these aren't visible unless you view two fish side by side or have looked at a lot of koi and goldfish. The difference in mouth is often visible when you feed them. Sometimes it will give you a chance to look for barbels, too.
|Physical differences in koi and goldfish (credit unknown)|
Can they interbreed?
Yes. When found in the same habitat (either in the same pond or in the wild), the two species will hybridize, but the hybrids are infertile. Often times these hybrids will be the natural brown, but some can come out a beautiful solid black. When these are found in the aquarium trade, they are often breeding mistakes.
|Genetically verified hybrid carp goldfish found in the wild (Jim Negus)|
I hope you now have a better understanding of the differences between goldfish and koi. These are two beautiful fish that deserve to be recognized for their own merits.