When Kagome urged Inuyasha to pull out the sword so's Sessh-boy loses his face, well..
it means so he loses his pride.
If, after all that sword searching, his vile half-brother born from a human is able to claim the sword as his
 it would be a loss to Sessh. How dare a hanyou such as him possess a sword from his own father?
Why must a hanyou earn such power? These thoughts surely must run through Sessh's mind.
They're half brothers... sibling rivalry couldn't be far off; Inu getting this famed sword...

And, more on Inuyasha's ambition of power; though I warn it to be fluffy:
You sure about not protecting her, Inu?
Don't you like this pure child even a little? I mean, who was that who screamed her name?
Besides, through all that, she has her 'sit' command connecting you to her.
Now, you have this humongous sword.
Thing is, it ain't gonna activate without a heart willing to protect.

And now, enter Kagome; a bright, cheerful girl willing to help.
One from our world to compare the harsh life in the world of demons. Oh, how she looks like Kikyou.
Could she truly be a reincarnation of a dead miko? She awakened Inuyasha.
She can sense the fragments of the Shikon jewel!
Oh, Inuyasha...had you only agreed to her request, and not argued,
Tessaiga's transformation would be known to you...
Ooh, and a thought: Remember the battle with Yura?
Inuyasha said his Fire Rat kimono to be stronger than any armour.
In a way, that's a hard translation, but excepting that..we've seen it  withstand fire, and have a little resistance to Sessh's whips.
Mmh, if one wants to know, the Robe of the Fire Rat is in a Japanese legend, where this
pretty girl had lots of suitors and made the ones that refused to leave her do really hard, impossible tasks, one of them fetching the Robe of the Fire Rat.