As I hinted above, these symmetries are useful: they make the task of doing Physics possible. But they also have observable consequences. Associated with each of these symmetries is what is called a conserved quantity: some physically measurable quantity that can not be created or destroyed, but only transferred from one object to another. You are familiar with them to some extent so it is easiest to give examples. Energy is the conserved quantity associated with the fact that the laws of physics don't change with time. The following is a good example. Earlier today I ate a chocolate bar which contained food energy. My body stored this food energy, and can use it to do work on a bean bag, by throwing it up in the air, and converting the food energy to energy of motion. As the bean bag flies upward, the gravitational field of the earth did work on it, converting the energy of motion to gravitational potential energy. At the top of its trajectory all the energy of motion is used up, so it can't go any further. This energy of motion is recovered as it falls, and then dissipates in the form of noise, etc when it hits the table.
Momentum is the physical quantity associated with the invariance of physical laws under translations in space. Momentum is essentially the propensity of a body to keep going in a straight line. Anybody who plays pool relies on the transfer of momentum from one billiard ball to another to sink a shot. A moving train has a great deal of momentum, and you can easily verify that this momentum cannot be destroyed, only transferred to other objects, by throwing yourself in front of the train to try and stop it (I don't recommend it though). Some momentum will be transferred to you, but not enough to slow the train down appreciably.
Angular Momentum is the conserved quantity that arises from the invariance of physical laws under rotations. It is the least familiar, but also the neatest. Angular momentum is the rotational analogue of ordinary momentum: the propensity of a rotating body to keep rotating unless it is able to transfer its rotational motion to some other body. This can be demonstrated by using a bicycle wheel. The counterintuitive behaviour of a gyroscope is explained by the fact that the angular momentum of the rotating flywheel in the gyroscope cannot be destroyed, and is difficult to transfer.
One important practical consequence of all this is that symmetry (applied to physical laws) is useful. The more symmetries a theory has, the more physical quantities that can be identified as conserved (neither created nor destroyed), and the easier it is to actually solve the mathematical equations and make predictions. It is partly of this reason that theories with more symmetries are thought to be more desirable. However, apart from this practical consideration, the presence of symmetries makes theories more beautiful in the sense that I tried to convey earlier: by possessing more symmetry, a given theory can incorporate, or synthesize more diverse phenomena into simpler structures. It is this feature that gives them at least part of their aesthetic value.