These two are synonymous although there are obvious differences in the way they manifest to the physical world or the environment.
First, inner strength are intangible qualities of a person, whereas personal power are demonstrable qualities of a person that are not only noticeable by society but are also authoritative qualities that may compel or persuade others to follow, emulate or imitate, admire, agree, etc. the powerful personality.
Next, inner strength is derived from the acknowledgement, acceptance, conviction, etc., of one's inherent qualities or the generic qualities of the person and his/her ability to stretch or enhance those good qualities to the point of excellence; while personal power is the natural effects to others based on the inner inner strength at work.
And last; an example:
Let's say in the inner strength I acknowledge, believe, see myself, convinced myself to the core as being beautiful, intelligent, with exemplary high moral/spiritual values, and rich. I am thus convinced of my inner strength because I did everything to enhance my qualities and had proven these qualities as true in my life experiences and in all my dealings with other people in different walks of life.
As to be convinced of being beautiful; I had done all my beauty and healthy regiments that result to most people I associate agreeing on my charisma, charm, beauty and to some extent, having many sincere and serious admirers and suitors to tail and desire for my approval.
My conviction of having high moral and spiritual values come from my strong family foundation and orientation; and from my four years in a religious Missionary training; hence despite the various temptations for promiscuity, immorality, etc., caused by many suitors, I have the wisdom to select only one lover and be the faithful wife to the best man I had selected.
My belief in my intelligence come from the fact that I studied hard and had tested my intellectual capacity as evidenced when I graduated Magna Cum Laude in a prestigious State University with the degree in Economics and more degrees in MBA, Education and Social Sciences.
As to the belief that I am rich; I am never in poverty because I got a noble employment in which the compensation goes well with it.
To sum up; inner strength is the acknowledged, and enhanced inherent qualities of a person; whereas personal power is the natural tendency resulting from the inner strength.