What are the differences between Cat I, Cat II, and Cat III as concerns aircraft requirements?
The differences are in the demonstrated accuracy of the ILS or RNAV equipment. Not sure if the question can be properly answered without writing a book, but digging into the documents linked by Jonathan Walters I extracted the tables below. The document is very long and full of caveats, exceptions and specifics on testing requirements, etc. and frequently refers to parameters set forth in other documents. 



CAT I



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CAT II



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CAT III


  Landing System Performance.   All types of low  visibility  landings systems, including  automatic flight control, guidance for manual control, and hybrid, shall be demonstrated to achieve  the performance accuracy  with the probabilities prescribed in this section.  The performance values may  vary  where justified by  the characteristics of the airplane. 
  
  The performance criteria and probabilities are as follows: 
  
  
  (a)  Longitudinal touch down earlier than a point on the runway  200 ft. (60m) from  the threshold to a probability  of 1 x 10-6; 
  (b)  Longitudinal touch down beyond 2700 ft.(823m) from  threshold to a probability  of 1 x 10-6; 
  (c)  Lateral touch down with the outboard landing  gear more than 70 ft. (21.3m) from  runway  centerline to a probability  of 1 x 10-6. 
  (d)  Structural limit load, to a probability  of 1 x 10-6.  An  acceptable means of establishing  that the structural limit load is not exceeded is to show  separately  and independently  that: 
  
  
  (i)  The limit load that results from  a sink  rate at touch down not greater than 10 f.p.s. or the limit rate of descent used for certification under 14 CFR part 25 subpart C (see section 25.473), whichever is the greater. 
  (ii)  The lateral side load does not exceed the limit value determined for the lateral drift landing condition defined in part 25, section 25.479(d)(2). 
  
  (e)  Bank  angle resulting  in hazard to the airplane to a probability  of 1 x 10-7.  A hazard to the airplane is interpreted to mean a bank  angle resulting  in any  part of the wing, high  lift device, or engine nacelle touching  the ground. 6.3.2.  Speed Control Performance.   Airspeed must be controllable to within +/- five  knots of the approach speed*, except for momentary  gusts, up to the point where the throttles are retarded to idle for landing.  For operations flown with manual control of approach speed, the flightcrew must be able to control speed to within +/- five  knots of the approach speed.
  


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