What is the main difference between an ETOPS certified and non-ETOPS certified plane?
The FAA ETOPS certification requires certain standards from both the airline (the operator) and the aircraft (the manufacturer). 

As far as the manufacturers are concerned, the ETOPS certification is given for a particular aircraft-engine combination satisfying certain requirements. The FAA ETOPS Rules Section A deals with the Airplane-Engine Type Design Approval. According to the document, the type approval involves a two step process:


  First, the FAA determines that airplane systems meet certain design standards for safe operations during an airplane diversion.


Some of these requirements are:


The aircraft should have at least three independent electrical generators.
The auxiliary power unit (APU) (when required) can start after the airplane has been cold-soaked for several hours and can run reliably for the remainder of the flight.
System safety analyses have to show that expected system failures will not prevent safe landing at a diversion airport. 
Systems with time-limited capabilities (typically cargo fire suppression) need to have the capacity to support a maximum-length diversion,
including a 15-minute allowance for a hold or go-around at the diversion
airport



  The second part of the approval process is an evaluation of engine in-
  flight shutdowns and other significant airplane system failures that have
  occurred while the airplane-engine combination has been in service.


The aircraft-engine combination should have at least 250,000 engine-hours for evaluation, though this can be reduced with compensating factors. The airplane-engine combination should maintain a target IFSD (In Flight Shut Down, defined as "When an engine ceases to function (when the airplane is airborne) and is shut down, however briefly, whether self induced, flightcrew initiated or caused by an external influence.") rate at or below 0.02 per 1,000 engine-hours for 180min ETOPS approval.

The manufacturer should also demonstrate that flying with OEI doesn't cause undue stresses to the aircrew and is safe for the airframe.

The ETOPS design approval has to be applied for with the FAA and obtained after the relevant processes. Thus, the same aircraft will not have ETOPS approval if the engine is different or even the same aircraft-engine combination won't have ETOPS approval if used by another airline or if no certification was sought.

There is another method of getting ETOPS approval called the 'early' ETOPS approval which 


  ...takes a systems approach to the development of an airplane and engine. ... an applicant must demonstrate that the design flaws on previously designed airplanes are not present in the new airplane. The applicant must also consider how the maximum length flight and diversion affect the design and function of airplane systems to ensure that they have the capability and reliability for safe ETOPS flight.


After being ETOPS certified, the airplane-engine combination and the operator are under continuous monitoring with the certificate being withdrawn in case of lapses or if the IFSD rate goes above the target rate and the manufacturer's corrective action does not eliminate the unsafe condition.

As far as the LAX-Hawaii incident is concerned, according to American Airlines Spokesman, both aircraft (the 'correct' ETOPS airplane, A321H and the wrong one sent, A321S) had the same configuration, but


  There are two small differences in terms of equipment to allow ETOPS certification. Since you can’t divert for medical [on a Hawaii bound flight], you’re required to carry extra medical oxygen on board and an extra fire suppression canister


There are also reports that the (ETOPS certified) A321H has extra fuel tanks, which the (non ETOPS certified) A321S didn't have, though the AA spokesperson had said that the fuel tanks and range were the same.