How differently will two bikes handle that have 1 degree difference in their head tube angle? I am looking to buy a new mountain bike and comparing two models with different head tube angles.  How big of difference does 1 degree of slackness make?  Would an intermediate rider be able to feel the difference?  

Both bikes appear to be fairly slack.  


The first bike has a 67 degree headtube angle.  
The second bike has a 68 degree headtube angle.
There is far more to the stability of a bike than just fork rake and head tube angles. I have designed bikes for a couple of years and the number of parameters is sometimes beyond believe. I have seen research that proved that a bike with a 90 degree head tube angle will stay upright by itself. Aha, you will say than it will be the centrifugal power from the wheels! Nope, the bike had two small skate wheels stacked on top of each other, so no influence there. It turned out that for this bike 'simply' altering the balance by shifting a weight made it stable : 
                
             or look for Two Mass Skate Bicycle on Google.

So to answer your question : you will not notice the difference in the headtube. What you will notice is the overall difference in handling between the two bikes coming from the sum of all components in the frame design. So go and test them to make sure you get the one that fits you best.