Rock, paper, scissors game in Python 3
First I suggest you to split your logic into several function, it will help you to see what is essential and what you could optimize / remove. 

import random
print("Rock, Paper, Scissors")

AVAILABLE_CHOICES = ['rock', 'paper', 'scissors']


def rps():
  try:
    userInput = getUserInput()
    computerChoice = getComputerChoise()
    result = getResult()
    tryAgainMaybe()
  except Exception as err:
    print(err)


  def getUserInput() :
    userInput = input("Please enter your choice: ")
    userInput = userInput.lower()
    if userInput not in AVAILABLE_CHOICES:
        raise Exception('Your choice is invalid')
    return userInput;

  def getComputerChoice():
    return random.choice(AVAILABLE_CHOICES);

  def getResult():
    result = ''

    if userInput == computerChoice:
      result = "DRAW!"
    elif userInput == 'rock'     and computerChoice == 'scissors' or \
         userInput == 'paper'    and computerChoice == 'rock' or \
         userInput == 'scissors' and computerChoice == 'paper':
        result = 'USER WINS!'
    else:
        result = "COMPUTER WINS!"
    return result

  def tryAgainMaybe():
    print("Would you like to try again?")
    try_again = input("Y/N  ")
    try_again = try_again.lower()
    if try_again == "y":
        print("\n")
        rps()
    else:
        print("\n")
        print("Thank you for using my Rock, Paper, Scissors Program")



Create a list of available choices instead of writing them everywhere. it will be easier to modify and to test for your input values.
random allow you to select ramdomly a value in a list.
you don't need to calculate every single possibility. You can either have a draw, a win or a loose. if you don't win or draw, you loose, no computation needed here.
when you name something, try to use the same name for the same thing everything, otherwise it could be missleading. eg (userInput / computerChoice) why do we have input and choice two different words for the same kind of value ? i would choose userChoice and computerChoice instead.
Your coding convention does not respect pep8 standard. You should be using snake_case.