How is this EEG circuit amplifying and filtering input correctly?
ADS1299 is a complete analog front-end, with cool opamps and ADC in one package. http://www.ti.com/lit/ds/symlink/ads1299.pdf.

The filtering is better to do in the digital domain, where you can apply any filter and you can easily have e.g. a 10th order of filter with a very steep cutoff curve. In contrast, the DIY solution uses a lots of analog circuit to get a simple 2nd order 60Hz filter - which makes a noticable attenutation over 20Hz.

The openBCI design is even more better, since it uses a floating bias on the patient, see the datasheet, and offers a 24 bit resolution.

Note that on the openBCI you may not see high pass filters (those are part of the ESD protection network), nor an instrumentation amplifier. Since the whole thing runs on batteries, it makes no sense to use an instrumentation amplifier. Instead of isolating the test leads with instrumentation amplifier, the openBCI solution isolates the whole device from the computer by using Bluetooth at once.

Talking about ESD, the DIY solution has no attempt to fix that - so it is basically really really sensitive and you can kill it with nylon clothing in no time :(

Oh, and finally, the openBCI solution is a true multi-channel solution, instead of the component hell of the DIY.