We tend to think older is better, but often do not define what we mean by better.
There are those ecologists who put great importance on biodiversity. Thus it is understandable they may assume mature forests are better and thus more bio diverse because of their age. This assumption appears very reasonable to those who think of a forest as a static ecosystem.
Forests are ecosystems, but they are not static. While changes may take decades or centuries, all forests are changing even if only replacing dying trees with new growth.
Forests pass through a series of stages as they age. Every now and then a catastrophe destroys a forest. That may be a forest fire, disease, volcanic eruption or a human activity such as agriculture. For example, If an efficient farmer clears land and grows crops, most of the forest species are eliminated from a large piece of land. If the farmer eventually abandons the farm, as happened in Northern Wisconsin at the end of the 19th century, the vacant fields will revert back to forest. What we have to understand is that the new forest is not the same as the one the farmer cleared.
The first plants to return to bare land are the pioneering species. They may not even be trees. Pioneering species include grasses, berry bushes and fast growing trees such as aspen and birch. These pioneering species share the following properties. Most of them can thrive on poor soil. They do not tolerate being shaded by other plants. They are relatively short lived plants that mature, die and rot within a short time. There are few aspen trees that are older than 60 years old. Aspens over 30 years often have core rot. Pioneering species generally have good propagation strategies through wind blown seeds and/or by sprouting roots. These pioneering species quickly take hold because they need a minimum to survive. As the forest of pioneering species ages, the tree species and larger bushes tend to displace the grasses and lower bushes.
The leaf litter of the pioneering species produce a rich layer of forest top soil. Trees also shade large areas of the ground.
Intermediate species, generally pines and spruce, Require a rich soil for the seedlings. Most intermediate species do better with only partial sunlight. Thus as a forest of pioneering species matures, the intermediate species have an advantage. Slowly over time, the intermediate species displace the pioneering species. The forest grows denser. There is less grass, a few bushes and much deeper shade than a forest of pioneering species. The forest topsoil continues to grow deeper and richer as leaves and needles fall.
Eventually, and this may take several centuries, the terminal species get a foothold. Terminal species such as oaks and maples are durable trees that endure. They grow slowly, but are strong and more rot resistant than pioneering species. Most of the seedlings of terminal species require a very rich soil and grow well in deep shade. Allowed to grow unmolested, the pioneering species and intermediate species will eventually be replaced leaving only terminal species. Some of these will be large and old. Under their canopy will be smaller, younger trees that are prepared to take the place of the old trees when they eventually die.
Depending on what kinds of trees a person values, most people look at an advanced intermediate forest full of pines or an forest full of oak trees in transition between an intermediate forest and a terminal forest, and call it a mature forest. In actuality, a mature forest is a forest consisting almost entirely of old terminal species.
In terms of plants, the forest was more bio diverse early in its life at the time when trees and shrubs were starting to displace grasses. From that point on, species are being displaced by other species as the forest progresses from one stage to another.
There is also a progression of animal life as food sources and cover change. Some browsing animals do not do well deep in an intermediate forest unless they can find clearings with pioneering species. Many of these browsing animals live either in forests of pioneering species or on the edges of forests where they can forage outside of the forest itself.
A forest is a a changing ecosystem. As important as biodiversity is, biodiversity is not a factor that increases as a forest ages. Allowed to age naturally over the centuries, a forest will undergo many changes, some of which will increase biodiversity for a while, but most of these changes ultimately reduce biodiversity.