Neshama

 לנשוף in Hebrew means to "blow", to breathe out, to exhale

לנשום

On the other hand means to breathe in, to inhale. 

The pasuk says: ויפח באפיו נשמת חיים , this means that God "blew" into man "נשמת חיים" The Neshama is the "Inhalation of Life" literally translated.

In the sefer Nefesh Hachaim it is explained in detail how the Neshama connects to man and how man's actions effect upwards and then back down. The Neshama is man's connection to God through man's actions. These actions connect through God's inhalation, the effect is from down to up. ויפח would connote an up to down relationship. Obviously the relationship is bilateral however, it is important to recognize when the connection is being created by man and when it is being created through up high. Besides for the initial creation of man when God breathes down it usually imports a negative connection. The right connection has to be initiated by man. Hence נשפת ברוחך כסמו ים was a terrible punishment meted out to the Egyptians.  We blow the שופר on Rosh Hashana to symbolize that we are initiating the connection and therefore praying for a positive response. We see in the Torah that when it says וירד it portends a punishment. Again it is God initiating the connection i.e. God coming down on his own rather than man climbing up towards him. Yaakov Aveinu's angels first went up the ladder then afterwards the angels came down. We pray and then and only then expect the shefa to come down from up high. The Bais Hamikdash is where we climb up close to Him and He then comes down towards us. This is all symbolized by the Neshama/Neshifa relationship.