A fear may mean that something upsets you slightly, while a phobia means that it terrifies you (as shown above). A phobia is a fear with a capital F. Fear of public speaking is a social fear. Speech anxiety is a type of social phobia.
One simple way to think about the difference is just to ask how much money you would be willing to pay to avoid giving a speech. Would it be:
A. $ 0.10
B. $ 100
C. $ 100,000
D. $ 100,000,000
E. $ 100,000,000,000
If you answered A or B, then you just have a fear. If you answered E, the hundred-billion-dollar ransom once demanded by Dr. Evil in an Austin Powers movie, then you definitely have a phobia.
A detailed clinical definition for a social phobia can be found on page 411 of the massive Diagnostic and Statistical Manual of Metal Disorders (4th edition, 1994), usually abbreviated as the DSM-IV, which begins by stating that:
“The essential feature of Social Phobia is a marked and persistent fear of social or performance situations in which embarrassment may occur (Criterion A). Exposure to the social or performance situation almost invariably provokes an immediate anxiety response (Criterion B). This response may take the form of a situationally bound or situationally disposed Panic Attack (see p. 394). Although adolescents and adults with this disorder recognize that their fear is excessive or unreasonable (Criterion C), this may not be the case with children. Most often, the social or performance situation is avoided, although it is sometimes endured with dread (Criterion D). The diagnosis is appropriate only if the avoidance, fear, or anxious anticipation of encountering the social or performance situation interferes significantly with the person’s daily routine, occupational functioning, or social life, or if the person is markedly distressed about having the phobia (Criterion E)....”
A few surveys have looked at how common both social fears and social phobias are. There was an article on “Social Fears and Social Phobia in the United States: Results from the National Comorbidity Survey Replication” by A. M. Ruscio et al. You can read it here.
The bar chart shown above (click on it to enlarge) shows that about 20% of adults in the US fear public speaking, while about half that, or 10% have a phobia. That 10% is much smaller than some of the nonsense you will find online, like at SpeakFreaks which instead claims that:
"Speech Anxiety is believed to be the single most common phobia affecting as much as 75% of the population, and as you have probably already heard, the fear is ranked higher than death!"