In short, they both have their own advantages.
A concrete pond is more permanent than a liner pond. This can be a real advantage if the pond is designed and built properly but a real nightmare if it is not. The inside contour can be built so the pond has smooth flowing walls leading from the top to the bottom drain. This can make the bottom easier to keep clean. Many professional pond builders prefer to build concrete ponds because they can draw professional help from the pool industry to install the rebar and gunite or shotcrete. Finding qualified people with a lot of experience installing liner ponds is hard. In general it is easier to install rockwork on a concrete pond compared to a liner pond if you want the rockwork mortared in place.
The disadvantage of a concrete pond is that they cost more to build. This added expense may be off set to some extent when you consider the value a well built pond may add to property value. Most bank appraisers consider a concrete pond similar to a concrete swimming pool, whereas, they consider a liner pond similar to a portable pool which has less value. An additional disadvantage of a concrete pond is that for the first three to five years they tend to have a higher than normal ph. If a high ph is of concern to you this can easily be overcome by letting the concrete cure and then coating the pond surface up to the water line with a rubber base paint or epoxy.
A 45 mil rubber liner pond is not as permanent, which can actually be a benefit. If the pond has some design flaws that you need to change or if you need to remove the pond, a rubber liner is much easier to work with. This is a big reason most “do it yourself” pond builders choose a liner pond over a concrete pond. A liner pond is also less expensive to build and a liner pond tends to maintain a more neutral ph. The rubber liner creates a very soft surface for the Koi but even when a liner pond is installed properly you will have a number of minor folds in the liner surface.
Once the pond is filled with water these folds become barely visible. Rock work around the edge of a liner pond can be a challenge if you don’t want the liner to show. The reason is because the rock work will need to extend down into the water and the mortar for the rock work will not stick to the liner. (To solve this problem see Advantage Pond Design Plans under Rock Work.)
The problem of cutting a hole in a liner for the bottom drains, jets, etc. is easily overcome with the proper drains and bulkhead fittings. When installed properly a liner pond can easily be leak free for many years.