I have been riding a road bike for 2 years now and I am being forced to replace it due to damage. I am now considering upgrading to a 11-28T rear cassette. Would doing so give me a major advantage as opposed to a 14/28T cassette?
Like another answer pointed out, there will be a difference in how the transition between gears feels; assuming they have the same number of sprockets, shifting from one gear to another on the 14-28 should feel "smoother" than on the 11-28.
From a performance standpoint, the 11-28 will give you the option of more speed, by virtue of you having smaller sprockets available.
However, you asked whether these things are noticeable, to which I would say: it depends. Here's why:
If you are a commuting or more recreational cyclist, the answer is likely no. Many commuters don't spend much (or any) time near the bottom of the cassette. If you're more of that style of rider, you may only get to the bottom of the cassette during a steep descent. Many of the commuting cyclists I know have about as much use for 11t cassettes as they do 53t chainrings.
On the other hand, if you are a more of a competitive or racing cyclist, you will most likely be able to notice the addition of the smaller sprockets; both on descents and flats, depending on ability. Some of the racing cyclists I know regularly find themselves near the bottom of their cassettes, during hard efforts. If you're a racing or aspiring racing cyclist, you'll probably notice the advantages of the 11-28 cassette.
With 11 teeth, you'll get more range when you max out. However, the 14-28 cassette will feel smoother as you go through the gears, because the number of teeth of adjacent gears will be that much closer.
For my money, you spend little enough time in those highest gears that any "advantage" is debateable.
Would doing so give me a major advantage as opposed to a 14/28T cassette?
I usually ride 50T/15T. If you have a 50T chainwheel too, this means the sprocket closest to 15 teeth may be 14-tooth sprocket. The chain will be at an extreme angle due to needing to use the highest gear (although the second highest gear may be 16T, which could be close enough to 15T).
In contrast, 11-28T cassette probably has something like 11-13-15-17-... meaning the sprocket closest to 15T is neither the largest sprocket nor the second largest sprocket.
Ideally the cassette should be selected so that its lowest and highest gear are for very rare situations, being almost completely unused in regular riding, and the rest of the gears are the commonly used gears.
A chain running straighter will have better lifetime and reduced friction.
You can somewhat reduce the problem by selecting 53T chainwheel instead of 50T chainwheel.