Javascript - Pick random array IF same as current array then pick another

I would define the array such that it gets initialized only once. Then I would select a random entry m between 0 and n-2 (with n the array size) and exchange the m-th entry with the n-1 entry and display this entry. Thus the new selection can never select the currently displayed entry.

One way to do this would be to compare the content of the element with the randomly selected new content and use a loop to select a new one until they are different.

var currentContent = document.getElementById("randomtestimonial").innerHTML;
do {
  rdmQuote = aquote[Math.floor(Math.random()*aquote.length)];
} while(currentContent == rdmQuote);
document.getElementById("randomtestimonial").innerHTML = rdmQuote;

This code could be improved since it's possible that if aquote.length were ever 1 then this would be a infinite loop. But, hopefully, it's a good starting point.

I would just cache the current selection.

  <!-- Hide the script from old browsers //-->
 var _cache="";
 function quotes(){
 //Define and populate the array 
 var aquote = new Array;
   // all your quotes

 //Generate a random number then print the quote from that array index
 rdmQuote = Math.floor(Math.random()*aquote.length);

 if (_cache == rdmQuote) 
     {
         alert('quote is same as current')
     }
     else
     {
       document.getElementById("randomtestimonial") .innerHTML=aquote[rdmQuote];
 }
 _cache  = rdmQuote;
 }

Just use a variable to store the current quote index.

var current = -1;

function quotes() {

    //Define and populate the array 
    var aquote = new Array;
    aquote[0] = "\"Your cakes are fantastic, beautiful designs and taste gorgeous!\"";
    aquote[1] = "\"I can’t believe how beautiful the cake was and how much detail there was on it.  My mum cried when she saw it and didn’t want to cut it up but we eventually persuaded her and it was really tasty.\" Sasha – Rothwell";
    aquote[2] = "\"Thank you for our wedding cake.  The fruit cake was absolutely delicious and so moist.  The flowers you made were beautiful and exactly as we imagined they would be.  We have kept the flowers and they are a great reminder of our wonderful day.\" Paul & Jane – Rutland"
    aquote[3] = "\"My husband said that the cupcakes you made for his birthday are the best he has tasted and your buttercream is divine – I have to agree!\" Dawn – Cambridgeshire"
    aquote[4] = "\"Thank you Bumble Cottage Cakes for My son’s birthday cake it was fantastic as usual I will be back soon and I can’t wait for the next one.\"Liz  – Desborough"

    //Generate a random number then print the quote from that array index
    rdmQuote = Math.floor(Math.random() * aquote.length);
    if (rdmQuote == current) {
        alert('quote is same as current')
    } else {
        document.getElementById("randomtestimonial").innerHTML = aquote[rdmQuote];
        current = rdmQuote;
    }
}

Your job is easier than your question title implies. You're looking to pick a random string from an array, and if it's the same as the current string, pick another.

You already know how to access this string, too, doing

rdmQuote = Math.floor(Math.random() * aquote.length);
if (rdmQuote == document.getElementById("randomtestimonial").innerHTML]) {
    alert('quote is same as current')
} else {
    document.getElementById("randomtestimonial").innerHTML = aquote[rdmQuote];
}

Note, I used a double equals, not the single you did. = is assignment. == and === are equality comparison. This still leaves you with the question of what to do if it is the same, (other than alerting). You need the do/while control command.

var quote = "";
do {
    rdmQuote = Math.floor(Math.random() * aquote.length);
    quote = aquote[rdmQuote];
} while (quote == document.getElementById("randomtestimonial").innerHTML;
document.getElementById("randomtestimonial").innerHTML = quote;