How do Aquinas and Hooker differ in their doctrine on communion? In The Screwtape Letters (1942), C. S. Lewis records the words of the devil Screwtape on church shopping and factionalism (letter 16):


  The real fun is working up hatred between those who say "mass" and those who say "holy communion" when neither party could possibly state the difference between, say, Hooker's doctrine and Thomas Aquinas', in any form which would hold water for five minutes.


What is the difference in doctrine on mass / communion, between Thomas Aquinas and Richard Hooker? 

Please ensure answers will hold water for five minutes.
Aquinas taught transubstantiation -- the Real Presence of Christ in the Eucharist:


  Some men accordingly, not paying heed to these things, have contended that Christ's body and blood are not in this sacrament except as in a sign, a thing to be rejected as heretical, since it is contrary to Christ's words...
  
  He is invisibly under the species of this sacrament, wherever this sacrament is performed...
  
  Christ is not really present in the other sacraments, as in this [the Eucharist]; and therefore the substance of the matter remains in the other sacraments, but not in this...
  
  But here the substance of the bread is changed into the body of Christ...
  
  Summa Theologica


Richard Hooker taught more along the lines of a pneumatic presence of Christ -- the elements are more than mere symbols, but the change takes place not in the elements, but is rather an act of God in the life of the believer:


  Eucharist is the sacrament of continued growth in Christ.  The Eucharistic elements are the bread and wine while the words are "this is my body," "this is my blood."  These words signify that the life of the body of Christ, the human nature, is present and active as grace in the Eucharist.  This grace flows from the Deity of Christ into his human nature, and from there, by the presence of the person of the risen Christ in his human and divine natures, the one fully present, the other present "after a sort."  This grace is then conveyed, as effects, to the bodies and souls of those who receive the bread and wine in faith.
  
  
    The bread and cup are his body and blood because they are causes instrumental upon the receipt whereof the participation of his body and blood ensueth.  For that which produceth any certain effect is not vainly nor improperly said to be that very effect whereunto it tendeth.  Every cause is in the effect which groweth from it.  Our souls and bodies quickened to eternal life are effects the cause whereof is the Person of Christ, his body and blood are the true wellspring out of which this life floweth (V.lxvii.5). 
  
  
  From this it follows that the change that takes place in Eucharist is not in the elements themselves, but rather, in the believer that receives the elements.  "The real presence of Christ's most blessed body and blood is not therefore to be sought for in the sacrament, but in the worthy receiver of the sacrament" (V.lxvii.6)...
  
  Hooker cannot believe that the sacraments are bare memorials, or merely remembrances of Christ.  Hooker believed the sacraments actually conveyed effects whose cause was the risen Christ.  
  
  
    For we take not baptism nor the eucharist for bare  resemblances or memorials of things absent, neither for naked signs and testimonies assuring us of grace received before, but (as they are indeed and in verity) for means effectual whereby God when we take the sacraments delivereth into our hands that grace available unto eternal life, which grace the sacraments represent or signify (V.lvii.5).
  
  
  Having rejected aspects of the Roman, Reformed, and Lutheran views, Hooker then observes that all parties agree on certain fundamental realities.  They agree on what God does in the Eucharist, though they disagree on how God does it.  The critical point, for Hooker, is what God does and not so much the how of God's action.
  
  (Robert Sanders, Richard Hooker on The Holy Eucharist)