What do the intra-Schengen border controls - reinstated in September 2015 - change?

Nothing much. Probable delay (and also general unpleasantness, especially if those checks involve some implicit “profiling”) is the main consequence for those who do have a visa or a right to travel in the Schengen area without visa. Also, what happened recently is that some countries decided to scale up and/or openly communicate their intent to reintroduce controls outside of any proper procedure. But internal border checks have been going on for some time, unofficially, and also occasionally been reinstated, completely officially, for special events.

All this probably violates several rules but that's something for the EU Commission to act upon. The Schengen agreement, implementation convention and regulations do not create any actionable right that would help travellers in a practical way if they face a control. Schengen rules also never officially included any provision allowing people to cross borders without ID, simply because carrying ID everywhere is mandatory in some Schengen states anyway.

Incidentally, this shows that the lift on internal border checks is only the tip of the Schengen iceberg and, perhaps surprisingly, the “softest” part of the system. If the EU and all member states continue to deliberately ignore it, the current situation could really go on for a long time without creating that much disruption. It would be much more difficult to ignore a member state which would stop honoring visas from other member states or issue many Schengen visas without following the rules (IIRC Italy briefly threatened to do the latter to force other member states to let asylum seekers go through their borders but never went through with it).

I passed through Amsterdam Airport several times during the period that the Netherlands suspended its Schengen commitments in March 2014. At the time I used Amsterdam as my principal hub because KLM was the only major carrier serving my local airport. The Schengen commitments were suspended because of the Nuclear Security Summit being held at the Hague.

http://nos.nl/artikel/616805-grenscontroles-terug-tijdens-top.html (Dutch)

The suspension made no difference to me at all, except once, when I was coming off a flight from Geneva: we all had our passports or ID cards examined on the jet bridge by Dutch police as we deplaned. It was as I remember a reasonably thorough inspection, but executed with a lot of efficiency.

It was mentioned on FlyerTalk: http://www.flyertalk.com/forum/klm-flying-dutchman/1558992-border-controls-all-arriving-passengers-march-14-28-a.html