Selective sexual reproduction and genetic engineering are actually markedly different. Genetic engineering involves precise splicing, insertion, or rearrangement of an organism's genome (or subset of genes). Selective sexual reproduction is a directed random rearrangement of genetic material over successive generations based (on often poorly understood) "meta-characteristics" or traits.
The critical distinction is that genetic engineering is the deliberate editing of exact genetic information, whereas selective sexual reproduction is a gradual, iterated, locally-random mixing of genetic information with imprecise results.
Furthermore, genetic engineering enables genetic mixing that aren't possible with selective sexual reproduction. For example, the insertion of genetic information into E.coli in order to produce human insulin for diabetics. Or, the modification of a particular cyanobacteria to secrete petroleum after photosynthesis.
http://pubs.rsc.org/en/content/articlelanding/2014/gc/c3gc42...
>> ...eliminating allergies, Alzheimer's, and cancer...
> That's all been doable for a long time, using controlled breeding...
This is incorrect. You may be surprised to learn that those deeply complicated, diverse families of afflictions would not be effectively treated via 'controlled breeding'.
A simple, naive disproof of the assertion that cancer can be eliminated in domesticated or selectively selectively bred animals (via artificial selection): pigs, huskies, and laboratory mice all get cancer at rates that are more or less congruent with wild boar, wolves, and rats.