A practical difference between the two sources can be seen in the case of a person who does most activities with his right hand but writes with his left. According the first reason, the left is still the weaker hand and would therefore be the correct one for the tefillin. The second reason maintains that writing is the primary consideration, so it follows that the tefillin would be laid on the left hand. Most Rishonim rule according to the first reason. A minority view which is followed by Rema rules according to the second reason. There are those who understand the second view as holding that both reasons should be taken into consideration, and therefore a person would not be considered a lefty regarding tefillin unless he does most of his activities--including writing--with his left hand.
[שו"ע כז, ו, משנ"ב כו-כז, וביה"ל ד"ה ויש, וד"ה והכי; ביאורים ומוספים דרשו, 25]