Perhaps the most exciting thing about these fish is their flashy colors. This appearance is what endeared these fish to people and brought about their transition from the farm pond to the garden pond. Both koi and goldfish exist because of selective breeding, thanks to the people in Asia who have been breeding the shiniest carp with each other for hundreds of years. The goldfish was here first; we know for sure that goldfish have been around for at least seventeen centuries, while the koi is relatively new here on the planet, having been around for just two centuries. The bright colors and sharp patterns of koi are their defining characteristic, and there are dozens of defined varieties, each with their own unique pattern of colors. Koi exhibit greater diversity of colors and patterns than goldfish, but goldfish come in a much greater variety of shapes.
Both koi and goldfish share the same habitat. They both like cool, fresh water that is relatively calm, with some plants, some shade, and a muddy bottom where they can root around among the pebbles searching for things to eat. Being smaller, goldfish can survive in smaller, more shallow ponds that would be inappropriate for koi. The recommended depth for an ecosystem pond with koi is at least 2', where goldfish can happily live in 16" if it never freezes. The area of a pond built for koi should be at least 8'x10', whereas a pond for goldfish can be half that size and still provide adequate room for the fish. If you are restricted in space, then goldfish, especially the smaller varieties such as shubunkin, are definitely a better choice.
The size, colors, and longevity of these fish are all subjective values, but one area where there is an objective difference between koi and goldfish is how much you spend on them. It is certainly possible to get koi on a goldfish budget, but in general, koi tend to be more expensive than goldfish. Individual goldfish rarely cost more than $40, usually just a fraction of that. Individual koi, on the other hand, may start at $5, but the price range goes much higher, often into the hundreds or thousands for large, top quality, healthy fish.
***While we're on the topic of introducing fish to your pond, this is a perfect time to stress the importance of properly quarantining any new fish you acquire. This prevents the introduction of parasites to your pond that could infect your healthy fish. Also, to prevent the spread of infection and parasites from a sick fish to a healthy pond, always get your fish from a reputable supplier. A good supplier will have knowledge of fish ailments and is doing their best to keep the fish they sell healthy.***
Both koi and goldfish eat the same food, but the amount they eat is relative to their size, so the larger koi will be eating much more than the little goldfish. They both will forage for food, and you can get away with not supplementing their food in an ecosystem pond if you're not concerned with allowing them to grow larger. The way we build ponds as an ecosystem allows nature to provide the food for your fish; this is one of the many benefits to an ecosystem pond. Many people are aware of this and still choose to feed their fish for other reasons: because it is fun, it builds a relationship between you and your fish, it can improve their colors (if given high-quality food), and in a water garden it may keep the fish from tearing up the aquatic plants such as water lilies in their search for food.
Where these fish differ on the issue of feeding is when the koi become much bigger than the goldfish. When young koi are introduced to a pond, their growth rate will depend not only on how much they eat, but on how much room they are given. If their pond is relatively small, their growth will be slower, as to not outgrow their home. There is a scenario where that may not apply, however, and that is when you move very large koi to a very small pond. In that situation, the fish were adapted to a larger environment where more food was available, and now they may require more food than their pond is producing. The bottom line here is it's possible to keep both koi and goldfish in an ecosystem pond without incurring any food expenses, and if you have six fish of the same size they'll eat the same amount whether they're koi or goldfish, but if you are feeding them yourself the koi will eventually grow larger than the goldfish and thus require more food.
Beyond these points, it is just a matter of preference. It has been said by koi enthusiasts that compared to goldfish, koi are smarter and have more developed personalities. They certainly have larger brains and more time to learn. For the most part, koi and goldfish are very similar to each other, and you can absolutely keep them together in the same pond. Most serious pond owners focus primarily on koi; as you might imagine, it is very rewarding to keep the same fish throughout your life and to establish relationships with them. Goldfish are a bit more aloof, and they are well suited to beginners and people who are 'just testing the waters.'
Many people never look back to goldfish once they've kept koi, but both fish make excellent pets to keep in your water garden, and there are scenarios that do favor one over the other. Provided your pond is well-suited to the fish you want to populate it with, follow your instincts and go with the ones that you find most appealing. You will be rewarded with years of enjoyment and companionship, and gain valuable insight into their underwater lives.