Of course it all depends on how the law is used and enforced. A couple of US based news sites opted to no longer be viewable in the EU because the companies apparently couldn't comply in time (though they had 2 years.) I doubt this will become more widespread, unless, again, complaints are filed and the law is enforced.
I've seen the opinion expressed that the law will hurt smaller entities more than the obvious targets which are the big corporations Hoovering up all our data. I tend to agree. They have the resources to stay ahead of, comply, and influence laws that smaller entities do not.
Specifically, I think independent content creators--individuals looking to monetize their content, most of all--could be stratified even lower in the internet's marketplace of ideas. Remaining independent simply costs more man-hours and money to remain compliant (to say nothing of responding to requests to pull/remove/rectify personal data), than it would to adopt an agency of some sort who can take care of all that for you, whether it is a publishing platform you don't own or control, or contributing to an actual publisher who ends up owning your content.
Also, it strikes me that free models for publishing platforms whose main source of revenue isn't selling data (think WordPress.com not Twitter) may become unfeasible. Ad revenues probably wouldn't be enough to offset it, either, as evidenced by all the major news sites that have moved to a subscription/read-limit model.
(eta: Actually, it quite reminds me of the whole weather widget fiasco, and then the explosion and stratification of the widgets/apps market.)
I could go on. The upside is that people (explicitly EU residents, I believe) gain new rights and control over whom, if anyone, has their data. I am not totally convinced of the practicalities of people exercising these rights, or that this will curtail the behaviors of the Big Data companies which brought this all about.
But at least your inbox should be quieter now.