The difference is in the rate at which the scaling happens. Exponential scaling will start off slower (Probably for a very short period of time, depending on your scale), but then get faster (Becoming much faster then linear scaling). In contrast, linear scaling will move at the same rate the entire time.

So, to have some real examples, let's say we're going to make a picture go from 100x100 pixels to 200x200 pixels in 1 second.

Linear scaling: ('seconds' is the amount of time that's gone by)

Our scale is 2 (Because at the end, our picture will be twice as big as it was. or in other terms 200 / 100 = 2) -- values of scale less then 1 will make our picture size decrease

'max_seconds' is the number of seconds the scaling will go on for, so 1 second in this case

Equation: new = old * (scale * (seconds / max_seconds))

0 seconds: 100x100 pixels

.25 seconds: 125x125 pixels

.50 seconds: 150x150 pixels

.75 seconds: 175x175 pixels

1 second: 200x200 pixels

Exponential scaling: ('seconds' is same as before)

scale is still 2

max_seconds is same as before

Equation: new = old * (scale ^ (seconds / max_seconds))

0 seconds: 100x100 pixels

.25 seconds: 119x119 pixels

.50 seconds: 141x141 pixels

.75 seconds: 168x168 pixels

1 seconds: 200x200 pixels

The difference between the two is that in the first, the change between each snapshot of time is 25 pixels. In contrast, with exponential, the change varies, changing slower at first (At .25 seconds, we're only gone 19 pixels from the start, vs 25 pixels for linear), but then increase (From .25 to .50, we change 22, from .50 to .75 we change 27, and from .75 to 1 we change 32). Notice how after .5 our changes are actually bigger then linear, this gives the appearance that it's moving faster at the end .

And a note if you know any differential Calculus (If not feel free to ignore this): The reason for the difference is that the derivative of a linear is a constant (And thus a constant change). The derivative of an polynomial (Such as x ^ (n) where n is a constant, Ex. our second case) is a line, or in other words a constant increase in our change.

-- Sorry, I realize this seemed like a lot of math blabber

but it does apply. If you stick to linear scaling them it should be smooth. The 'smooth-ness' really depends a lot on what number you pick for your intervals for the 'seconds' variable though (Or Ex. your FPS). A high FPS (And thus 'seconds' variable) results in a very smooth motion, where as the example I gave (With only 4 FPS) would be very jerky.

Matt