Manga quote (courtesy of Meguri): "Mikagami is magnificent. A golden valued genius swordsman. Kai... compared to him, you are only a stone!"
I'll probably expand this summary later. For now I only wanted to quickly type the basic differences. A few key notes are that Kai's Hyouma En (actually Meguri's) is a talking vampiric weapon, always howling for Mikagami's blood (it would have a particular taste for his blood, because he was trained with Ensui), and that Kai seemed to be going a bit bonkers from the chronic gold-stone comparison he suffered through. Poor guy! This leads to a nice thoughtful response from Mikagami, one that makes Yanagi cry. "Kai... don't mistake me. I am not gold. I only want to avenge my sister, that is the only reason I learned Hyomon Ken. That is the only reason, my desire for blood. I'm the same. To live alone in humiliation or to take the sword for revenge, what's the difference. Rather... your pure search for strength, it dazzles me."
In the anime, Mikagami uses up all of the water in Ensui to make the Tsurara Hebi. Then he looks like he thinks "Oops, gosh" and needs to get a little desperate. So he makes the blood Ensui, charges, yadda yadda. Gets stabbed. Goes down. All because he had a realisation.
Manga Mikagami is considerably more shrewd. He's had the realisation about Kai not being Mifuyu's killer, but he still needs to get the facts one way or another (this attitude has led to more pain in a recent volume). During the earlier parts of the fight you could easily say he is playing around too much, because he needs to figure out how to get his information. Finally, he knows he needs to talk to Meguri. It seems to him that his master had multiple personalities at best. There was the one who was a kind and loving second father to him. And there was the one who told Kai that he needs to kill Mikagami to become the successor of Hyomon Ken.
At this point, Mikagami knew he needed to ensure that he would be able to have a chat with Meguri. So... 1. He had allowed Kai to injure him. His own blood was necessary. 2. As a pure bluff, unlike in the anime, he used the Tsurara Hebi. The point in this was not to waste all of his water on one attack and hope for the best. It was to show Kai that he had used up all of his water, so he could catch him off guard.
Kai was fooled. He attacked Mikagami, supposedly hit him... but it was another water puppet (made out of blood this time, as you do see in the anime when it melts into a puddle of red Jello-like something). And then Mikagami stabbed him with the Ensui made out of blood.
All of this related to one of Kai's memories of Meguri gushing about the wonders of Mikagami. He'd told him that stone can become stronger than gold, but that even at his dying moment Mikagami would turn it all to cold calculation. And so he did. By getting injured so he could use his blood, by "wasting" his water on the Tsurara Hebi, by draining his own blood with Ensui and using it to create the blood puppet and the new sword to stab Kai with. That's a lot of blood! Even Kage was disturbed by the tactic!
Mikagami had drained enough of his blood that he fell down with Kai, but at that moment he only really cared about living long enough to confront Meguri.
If you're curious about more of the differences (they are immense) between Manga Mikagami and Anime Mikagami, I have a very long summary of his debut manga chapters up at the Midorikawa Hikaru Group under Files.