`Koh Keli is a poem/prayer which is made specifically as an introduction to Ashrei. That is why it is said before Ashrei.
You can tell this is so, because each of the paragraphs of Koh Keli end with "...Forever I will praise you with Ashrei Yoshvei Beysecha." The poem asks that we all be returned to our land and be able to offer the proper korbanos etc. Ashrei's first words are "Happy are those who dwell in your house..". So it is appropriate to precede Ashrei with a request that "Your house" be rebuilt so we can all return there etc.
Koh Keli is a "pre-mussaf" prayer because Ashrei is the usual pre-mussaf prayer/intro and Koh Keli introduces Ashrei. It was intended to get the community to keenly feel the lack of the Temple and the Land of Israel which is the only place to offer the real musaf korbanos. The one who wrote the piyut seemed to want the congregation to cultivate this in mind during the words of Ashrei so they would already have such a feeling flaming in their heart as Musaf started.
Hinneni he'ani... means "Behold I am a poor man...". It is the Chazzan's job to be the people's representative to lead the Musaf prayer. This job starts with the Kaddish before Musaf (which is after Ashrei). You need a Chazzan to recite Kaddish and everyone answers. Ashrei doesn't need a Chazzan.
So, right before the Chazzan begins his job, he declares how unworthy he is and begs Hashem that his and the congregation's prayers are accepted etc.
That's the difference. :)