Aaron and redfiloux both have great answers here. I'd like to try to address the side of "these are basically cohomology theories" a little bit more carefully, hopefully without being too off-topic.
As an aside, a great reason to care about cohomology theories comes from Adams' two solutions to the Hopf invariant one problem. His first solution uses ordinary cohomology to do the work, but it is very long and makes extensive use of "higher cohomology operations." His second proof (with Atiyah) is beautiful and short, but only because he uses an extraordinary cohomology theory (complex K-theory) to do the job. (http://people.virginia.edu/~mah7cd/Foundations/Adams,%20Atiyah%20-%20K-theory%20and%20the%20Hopf%20Invariant.pdf)
So suppose we care about cohomology theories. These are fairly simple on their own, they just turn each space into a sequence of abelian groups. Certainly we can get pretty far by just calculating these groups and using them to prove theorems.
Only for a given theorem, we may need to construct a new cohomology theory out of older ones. For instance, we may need to take a pushout or direct limit of cohomology theories. This is actually quite difficult if you think of cohomology as a functor from spaces to groups. You can make a category of such functors, but this category does not contain all colimits, so you are left without a means of "gluing together" cohomology theories to make new ones.
You may be familiar with a similar problem at the space level. One can construct the "homotopy category of spaces" by taking the objects to be CW complexes and the morphisms to be homotopy classes of continuous maps. This is a category, all right, but it does not contain pushouts or sequential colimits. So you can work with spaces and maps-up-to-homotopy if you like, but you won't be able to do much. It's much better to work with spaces and maps on-the-nose, and to make constructions like the double mapping cylinder and the mapping telescope when you want to form pushouts or sequential colimits. A good mantra is, pass to homotopy as late as possible.
The same reasoning applies to spectra vs. cohomology theories. The stable homotopy category of spectra (as described by Adams and Switzer) is almost equivalent to the category of cohomology theories (but not quite: https://mathoverflow.net/questions/117684/are-spectra-really-the-same-as-cohomology-theories). We can take double mapping cylinders (and homotopy cofibers, etc.) of spectra, and this gives us a meaningful way of "gluing together cohomology theories." The extra topology involved (a sequence of spaces instead of a sequence of groups) gives us more control and a greater ability to make constructions like this.
Hopefully that provides some motivation to study spectra!