Why are groups more important than semigroups?
To add a remark related to Jim Belk's answer and the OP's comments on that answer:
In many naturally occurring situations, including some of those where group theory is particularly useful, endomorphisms are automatically automorphisms.
For example, if $E/F$ is a finite extension of fields, any endomorphism of $E$ which is the identity on $F$ is automatically an automorphism of $E$.
As another example, if $C$ is a Riemann surface of genus at least $2$, then any (nonconstant) endomorphism of $C$ is necessarily an automorphism.
Any endomorphism of a Euclidean space which preserves lengths is necessarily an automorphism.
Another point to bear in mind is that the groups that arise in practice in geometry are often Lie groups (i.e. have a compatible topological, even smooth manifold, structure). One can define a more general notion of Lie semigroup, but if your Lie semigroup has an identity (so is a Lie monoid) and the semigroup structure is non-degenerate in some n.h. of the identity, then Lie semigroup will automatically be a Lie group (at least in a n.h. of the identity). A related remark: in the definition of a formal group, there is no need to include an explicit axiom about the existence of inverses.
To make a point related to Qiaochu Yuan's answer: in some contexts semigroups do appear naturally.
For example, studying the rings of endomorphisms of an object is a very common technique in lots of areas of mathematics. (E.g., just to make a connection to my first point, for genus $1$ Riemann surfaces, there can be endomorphisms that aren't automorphisms, but then genus $1$ Riemann surfaces can also be naturally made into abelian groups --- so-called elliptic curves --- and there is a whole theory, the theory of complex multiplication, devoted to studying their endomorphisms rings.)
As another example, any ring of char. $p > 0$ has a Frobenius endomorphism, which is not an automorphism in general; but the semigroup of endomorphisms that it generates is typically an important thing to consider in char. $p$ algebra and geometry. (Of course, this semigroup is just a quotient of $\mathbb N$.)
One thing to bear in mind is what you hope to achieve by considering the group/semigroup of automorphisms/endomorphisms.
A typical advantage of groups is that they admit a surprisingly rigid theory (e.g. semisimple Lie groups can be completely classified; finite simple groups can be completely classified), and so if you discover a group lurking in your particular mathematical context, it might be an already well-known object, or at least there might be a lot of known theory that you can apply to it to obtain greater insight into your particular situation.
Semigroups are much less rigid, and there is often correspondingly less that can be leveraged out of discovering a semigroup lurking in your particular context. But this is not always true; rings are certainly well-studied, and the appearance of a given ring in some context can often be leveraged to much advantage.
A dynamical system involving just one process can be thought of as an action of the semigroup $\mathbb N$. Here there is not that much to be obtained from the general theory of semigroups, but this is a frequently studied context. (Just to give a perhaps non-standard example, the Frobenius endomorphism of a char. $p$ ring is such a dynamical system.) But, in such contexts, precisely because general semigroup theory doesn't help much, the tools used will be different.
E.g. in topology, the Lefschetz fixed point theorem is a typical tool that is used to study an endomorphism of (i.e. discrete dynamical system on) a topological space. Interestingly, the same formula is used to study the action of Frobenius in char. $p$ geometry (see the Weil conjectures). So even in contexts such as action of the semigroup $\mathbb N$, there is some coherent philosophy that can be discerned --- it is just that it is supplied by topology rather than algebra, since the algebra doesn't have all that much to say.
I think the conclusion to be drawn is not to be too doctrinaire, and to be sensitive to the actual mathematical contexts in which and from which the various notions of group, semigroup, automorphism, and endomorphism arise and have arisen.
It is true that groups are much more important in mathematics than semigroups. There are two basic reasons for this:
Groups are the mathematical embodiment the concept of symmetry. In the examples you give, you seem to equate the idea of symmetry with automorphisms of algebraic objects. However, most symmetries that mathematicians care about are geometric or analytic, such as the rigid symmetries of a polyhedron, the deck transformations of a covering space, the automorphisms of a graph, the continuous symmetries of a system of differental equations, the isometries of a metric space, and so forth. In these cases, there isn't an obvious analogue of "endomorphism" that could be used to form a semigroup. Even if there were, the semigroup wouldn't be as useful, because the most interesting endomorphisms are the automorphisms.
Algebraic objects tend to have many more endomorphisms than automorphisms, which makes it much easier to understand the automorphism group than the endomorphism semigroup. Moreover, the definition of a group is "rigid" enough that it leads to a rich structure theory, which makes it much easier to investigate the structure of a given group than the structure of a given semigroup. Indeed, one of the first things you would want to understand for a semigroup would be the structure of the group of units.
Of these two reasons, I would say that the first explains why groups are so much more important to mathematics in general, while the second helps to explain why groups are more important even within the context of abstract algebra.
I disagree that groups are more important than semigroups. For example, the multiplication on a ring without identity turns it into a semigroup, and rings are incredibly important in mathematics. In fact a ring without identity is nothing more than a semigroup internal to the category of abelian groups with tensor product.
What I do believe to be the case is that groups were the first to be historically studied because it is more natural to think about isomorphisms than endomorphisms (it is not at all obvious from the perspective of our mathematical ancestors that a non-isomorphism is a useful thing to think about) and that groups are easier to study since they have more structure (e.g. the representation theory of finite groups is much easier than that of monoids or semigroups; see this question).