The Rama says (OC 38:3) that one must protest a woman's putting on Tefillin. The Magen Avraham there (sk 3) says that this is...:
מפני שצריכין גוף נקי ונשים אינם זריזות להזהר...
...because [Tefillin] require a clean body, and women aren't quick to be careful [about this].
What kind of cleanliness are we talking about? It's kind of hard to believe that women (even 400 years ago) would need to go to the bathroom uncontrollably.
The Aruch Hashulchan 38:6 writes that women are patur and if they want to we should protest because of the reason of having a clean body.He brings Elisha from the gemara in shabbas with the story of the Romans and his teffilin turning into kanfei yona ,the point was that Elisha was very careful in being clean while donning teffilin.One has to be in his state of cleanliness(breaking wind,falling asleep is forbidden).
So how can mean wear it they aren't any cleaner.So he answers men only wear it for prayer and we wear it because we are obligated but one who is exempt should not put themselves into a situation when it is very hard to be clean like Elisha.
He mentions that by Michal the chachamim did not protest because it was known she was a complete tzaddeikas and she was very careful.
