In western thought we consider someone who has reached their 1st birthday is 1 year old - i.e. we round down. So on June 1 2014 your person will be considered to be 30 years old, and until then he is 29 years old.
In some cultures, a part of a unit is considered one unit (i.e. they round up), and calendar years are used. For example, for a person born in December 2000, they will be described as being in their 2nd year through 2001, and in 2002 will be in their 3rd year. [citation needed]
(current year - birth year) evaluates to the age of the person in the current calendar year, i.e. the age they will be on their birthday this year. You might call this their "age this year" as opposed to their "current age", but I don't think these terms are widely used or would necessarily be understood by everyone this way.