In its simplest form, the difference is "variance drag", ie how volatility itself affects your mu above.
Imagine a series of random returns of +50% versus -50% with a 50% probability of each. Evidently this will have a mu of zero, and a sigma of 0.5.
But if prices halve and appreciate by 50% with equal probability, they will clearly decline over time. True "Mu" is not really zero. Estimates of linear volatility are thus biased.
On all the conventional assumptions of normality, the difference between the two mu's will be half the (linear/conventional) variance. So if linear mu and sigma are say 0% and 10% respectively, then you should expect the asset to depreciate by 50bps a year. Equally, if you always held a fixed amount of this asset (and did not let your holding in it compound, appreciate or depreciate), then this zero-return asset generate positive returns of the same 50bps.
If you want to understand this effect more intuitively, imagine if it doubled or went to zero with equal chance (instead of +/-50%). What then would be your expectations? Higher vol (eventually) guarantees gambler's ruin.
So market participants often just default to looking at everything in log terms (as per your textbook) simply to get around the distortive effects of sticking with the standard linear approach, taught to students in traditional statistics courses.
In most practical cases, linear and log vol will be almost identical to each other. The implication is more normally (no pun intended) the differing impact of this volatility on linear (arithmetic) versus log (geometric) averages [and resulting wealth outcomes!!!]