I'm not terribly clear on where George Borjas sees the data showing small effects from both immigration and minimum wage increases conflicting.
The minimum wage data shows minor or nonexistent levels of employment dislocation because, assumedly, employment is more inelastic than we'd expect, and it's rather costly for companies to operate with fewer low-wage employees. So when you raise the wage, the companies pay it, because the workers are worth more to them than $5.15 an hour. They just liked being able to pay that little.
The immigration data showing a relatively minor effect on wages suggests that immigrants rarely compete directly with native workers (that they are "complementary," in econo-speak), and so their entrance into the labor pool does not mean there's more labor supply for the same jobs, but also more jobs that simply wouldn't exist otherwise, thus reducing the impact on wages. (If there weren't Chinese immigrants, for instance, we wouldn't just have a bunch of Alabamians running the Chinese restaurants. There would simply be fewer Chinese restaurants, and people who liked Chinese food would cook more often.)
It's not clear to me that these two findings contradict. I'm no economist, but it seems that comparing the effects of a wage increase among all low-income workers with the effects of increased labor supplies among some low-wage industries that already have fairly low-native participation aren't in obvious tension. But I could be missing something.
Meanwhile, Giovanni Peri, who's actually conducted some of the research showing immigration has low wage suppression effects, explains his results this way:
On first thought, it might seem that the simple economics of supply and demand would answer the question: What is the effect of immigrants on wages? Immigrants increase the supply of labor. Hence, they should decrease the wages of native workers, reduce their employment opportunities, or push them to other states. The question, however, is more subtle than this, because all workers are not the same: They differ by education, skills, and occupation and perform jobs and productive tasks different from and complementary...
In nontechnical terms, the wages of native workers could increase because the increased supply of migrants is likely to put native workers in jobs where they perform supervisory, managerial, training, and in general interactive and coordinating tasks, which makes them more productive. Moreover, the presence of new workers also implies higher demand for consumption, so that immigration might simply increase total production and demand without depressing wages.
See Bruce Webb's comments, too.