There is considerable overlap between these modalities; the physiological changes that occur are very similar, however, the effects differ slightly.
From an anatomical point of view:
Hypertrophy training is the only modality that stands out when it comes to an increase in the muscle cross sectional area. Training power (slightly) and strength (more so) can give a stimulus high enough for hypertrophy to occur. The main difference between them and hypertrophy is the time under tension. Since the sets last a shorter period of time, there is less of a metabolic demand on the muscle. As far as I've seen (sort of like an average from different studies), low volume/high intensity training (power and strength) leads to about a third of the hypertrophy that you get from doing medium volume/medium intensity (hypertrophy) training. Since the effect most probably isn't from metabolic stress, I guess it can be due to the fact that the high tension in the muscles (more so from eccentric training) acts as a stimulus for satellite cells to proliferate and differentiate. By doing that, the muscle cells get a higher number of nuclei (and RER/ribosomes), which leads to a higher protein synthesis rate. Satellite cells are very interesting when it comes to muscle physiology, but to explain everything would be outside the scope of this question (I'll post a question about it and answer it for those that are curious: What is the function of myosatellite cells, and how do they relate to performance?).
Another aspect, that could be mentioned under the anatomical point of view, is that of toning. There are many misconceptions about toning in the general population. I'm not sure if there is an accepted definition (it probably depends on if you ask women's fitness magazines or athletes), but my definition would be that of an increased reflex arc. That is, the resting tension of the muscle is higher, due to a greater stimulation of the muscles through the reflex arc, which gives them a firmer appearance (from an esthetical point of view) and faster response rate (from a functional point of view). Contrary to women's magazines that say that one increases tonus by doing hundreds of reps at low weight, performing high intensity power-based activities increase the resting state firing of the reflex arc, giving firmer muscles.
From a histological point of view:
Strength, power and speed training (being high intensity, but of varying volume), all induce a fiber type change toward fast-twitch fibers. This effect takes a longer time and is of a lower magnitude than that of the opposite; a fiber-type switch from fast to slow. Generally, muscle fibers more easily assume a slow-twitch configuration, and the effect is more chronic (once your fast-twitch fibers become slow-twitch, it is hard or near impossible to get them back to the original fast-twitch percentage). To relate this to the anatomical point of view, having a higher percentage of fast-twitch fibers (because they hypertrophy more than slow-twitch) allows for a greater cross-sectional area of the muscle.
Hypertrophy training on the other hand induces a fast-to slow switch (because of the slower speed of repetition and longer time under tension). Body builders (which are the ones who spend the most time doing pure hypertrophy training) have a small percentage of fast-twitch fibers (about 35-40% on average), compared with powerlifters and sprinters (75-80%), and almost as low as marathon runners (20-30%).
Endurance training, as can be seen in the above example, causes the greatest fast-to-slow switch, for reasons that are self-evident. Another histological aspect of endurance training is that of capillarization.
Endurance training, which utilizes slow-twitch fibers, relies heavily on oxygen and as such require a larger capillary network. That is the reason why this type of training is the only one (with some overlap with hypertrophy training, depending on the set duration) to have a vasogenic effect.
Roughly, we can say that the muscle that trains intensively (high tension or rate of force development) becomes fast-twitch, and also increases its levels of creatine phosphokinase and glycolytic enzymes, while a mucle that exercises for longer periods of time become slow-twitch and have higher levels of mitochondria and myoglobin etc.
From a physiological point of view:
Strength, power and speed all more or less have the same effect on the output of alpha motor neurons; the frequency of firing increases and motor units grow (each motor neuron innervates a larger number of muscle fibers). These training modalities can all be classified under neural adaptation training, which is further explained here: Neural Adaptation Training vs. Hypertrophy Training?.
Hypertrophy training, and endurance training more so, have the opposite effects. That is because a large portion of the muscle remains at rest (lower tension) and slow-twitch fibers are specifically activated (they are part of smaller motor units and have a lower force production).
There are several more aspects which can be defined within the "physiology", such as VO2Max and resting heart rate, bone formation and strengthening, blood pressure changes etc., but explaining all those aspects would require a couple of more pages, and they are not as interesting when it comes to fitness and performance as the aspects I have explained above. If there is anything I've forgotten, please comment and I will add it to the answer.
