It depends what you understand each term to mean. Isn't it just semantics between words?
Unfortunately not. If you understand that dominance is a term that describes the personality of a dog and
assertive is a collection of behaviors that described how a dog is behaving them you are going to interpret the dog's actions differently and respond differently. And a dog's social structure, just like humans, is not that simple.
Many people associate the term dominance with a hierarchical social structure, one that relies on a vertical hiearchy. Their understanding is there is an "alpha" dog and a "beta" dog. The "alpha" dog is dominant over the "beta" dog for all things in life and so on down to an "omega" dog at the bottom of the hierarchy. The same way the human military is structured. General, Colonel, Captain down to Private. The lower ranks must abide by the rules of the higher ones, even if they are arbitrary, as the assumption is that the higher ranks know what is best for the entire group. Military structure exists in situations where one group of people band together to prevent others from taking something away, like during war.
With this way of thinking, the dog is always trying to rise up a social ladder and get control. And therefore you always have to be on guard because the dog is always trying to get a higher social standing than other dogs or their human. The tendency towards using violence in such a social structure is common due to high arousal. They have to be to be on guard at all times for the possible challenge from individuals lower down in the hierarchy.
War is a very extreme behavior among the same species and not normal behavior for animals at all. It occurs in less than 5% of normal social situations. Hence 95% of the time, the social mammals go out of their way to avoid conflict and have developed and learn from others very intricate behaviors to avoid damaging conflict.
Maintaining possession of a territory (piece of land) as an example is generally done is many subtle ways long before any physical interaction occurs.
The very regular presence of an animal in an area, using chemical communication like droppings and urine placed at regular intervals keeps other members of the same species from lingering. Even if they see each other at a distance, there is much posturing done.The posturing allows them to assess each other without risking injury. The less assertive animal can simply move away. As they get closer to each other or accidentally find themselves too close for comfort, there are more subtle behaviors that keep the peace and tell each other there is no need for alarm. "I am no threat." And during close social contact, there are still other behaviors that can help smooth over social fauxpauxs or misinterpretations.
Typically, when arousal level is high is when we see animals acting "dominant" in a situation over a resource. The arousal may be triggered by the fear of loss of the resource (or in humans feeling inferior, embarrassed and other emotions). It is often these very emotions and the chemicals triggered by them that can lead to aggressive behaviors and a loss of social self-control.
Demonstrations of dominance behavior and aggressive tendencies tend to be more common among non-social species who rarely interact with others of the same species. In general, they have higher overall levels of chemicals in their bodies related to aggression (vasopressin, testosterone) and lower levels of social calming chemicals (serotonin, and oxytocin etc).
During mating season is the most common place to see dominance behavior in mammals. The chemical communication draws in several males who find themselves in close proximity wanting the same resource. Arousal levels are high due to the potential for mating with the female. The female is a resource that needs to be protected by the male that is most capable of keeping the other males away. More chemical communication. Remember that mating occurs only once a year in most medium and larger wild mammals (with the exception of rodents, rabbits and some marsupials who can breed several times a year, providing the right conditions exist.)
The level of testosterone can affect how individuals relate to each other. We know that when we see the effect of steriods on weightlifters.
A dog that is perceived as assertive displays a collection of behaviors that show he is standing up for his own welfare without being aggressive with other dogs. He is only responding to a real threat and tends not to respond to all potential loss of a resource. For example, a dog that allows another smaller or younger dog to eat food out of his dish but who could easily win during a conflict between the two. Dogs that are confident in themselves do not need to show threatening displays (like a high tail or chin over shoulder) to get what they want. They use calm body language and respond to other's communication. They are often seen as the amicable ones.
Recent science has highlighted that dominance exists but occurs when one dog values one resource more than another dog and is willing to take risks to keep that resource. So for example, a dog that guards his food from another dog highly values that food and is willing to risk injury to keep it. A dog that protects his sleeping space is being dominant in that situation. The interesting thing is that functional well-socialized dogs have fluid social structure where they try to avoid conflict because conflict is dangerous at a biological level. They may start out resource guarding against another dog, but when they learn to trust that the other dog doesn't want the food as much as they do and are willing to respect the first dog's need for the food (and will have their needs taken care by their humans) then they no longer feel the need to protect it against that dog.
Dogs that exhibit true dominance are actually worried about losing something. If you have a dog that is demonstrating he is willing to take a risk in getting hurt for many things, you actually have an insecure dog, poorly socialized dog or a traumatized dog. Dogs with these characteristics do not make good service or assistance dogs.