Q:

Why do Groups and Abelian Groups feel so different?

Groups are naturally "the symmetries of an object".  To me, the group axioms are just a way of codifying what the symmetries of an object can be so we can study it abstractly.
However, this heuristic breaks down in the case of many abelian groups.  Abelian groups more often arise as a "receptacle for addition".  What I mean is that they are more intuitively counting combinations of some generating elements.  See for example: solution spaces to linear equations, the underlying group of rings and modules, (co)homology groups.  They rarely act naturally on anything except themselves, which seems like a copout.
It bugs me that these two intuitions are so far from each other, even though the underlying axioms differ by a single, deceptively-mild assumption.  Is there a way to reconcile these two perspectives?  Do abelian groups satisfy the group axioms by accident?

A:

Here's how I think about it: if the automorphism group of an object is abelian, this means something very strong about the object.  It sort of means you can affect its structure in two different ways, in any order, and they won't affect each other.  This to me hints that maybe the object consists of separate "pieces" that can only be affected independently.
The first example like this which comes to mind is a direct (commutative ring) product of finite fields of different characteristics.  Each field has a cyclic automorphism group, making it very "simple", and the fields can't map into each other, meaning they can only be affected "independently."
So I'd posit that if you want to think of abelian groups as symmetry groups, you can imagine them as the symmetry groups of objects that "break apart" into "simple" pieces which can't interact with each other (intentionally vague, since I don't want to commit to a particular category).

A:

From a representation theory perspective, abelian groups are distinguished by the richness of their invertible representations, since (in the locally compact case), the groups are completely determined by one dimensional complex characters.  For nonabelian groups, the one dimensional characters only determine the abelianization, and one must find actions on larger vector spaces to recover the full group structure.  Abelian groups admit a straightforward Fourier transform, which is a change of basis between functions on a group and functions on the dual (i.e., character) group.  For a nonabelian group, the situation becomes substantially more complicated, involving representations of quantum doubles.
From a homotopy theory perspective, groups can be delooped once, but abelian groups can be delooped infinitely many times.  This is why you can have Eilenberg-MacLane spaces K(G,n) for n > 1 if and only if G is abelian.  Since these spaces represent cohomology with G coefficients, this is why higher degree cohomology only exists for G abelian.
There are intermediate levels of loopiness, corresponding to commutativity up to homotopy plus some intermediate levels of coherence of choices of homotopies.  The quantum double of a group G, for example, lives on the two-fold loop space G/G.  A discrete two-fold loop space H is automatically an abelian group, since \pi2 of the two-fold delooping is H, and higher homotopy groups are abelian (this is one form of the Eckmann-Hilton argument).  As Qiaochu mentioned, the 2-morphisms in a 2-groupoid with one object and one 1-morphism form a natural two-fold "loop set" which therefore is an abelian group.

A:

I also like to think of abelian groups as quite different from (general) groups. To me an abelian group should be thought of as being more like a module than a group. Indeed, abelian groups are the same thing as Z-modules and the category of abelian groups is an abelian category (which has many nice properties not shared by the category of groups). The prototypical examples of abelian categories are categories of modules over a ring. In the case of abelian groups the ring is Z (the ring of integers).

