There are no disadvantages, except that for most of the series the Tenseiga cannot be used as a weapon. When Sesshomaru gains the Meido Zangetsuha technique, he becomes very powerful. However, he reluctantly turns it over to Inuyasha, after Inuyasha fights him in one last battle, and also since he [Sesshomaru] knows it was his father's will that the refined technique be given back to the Tetsusaiga. After that, the Tenseiga goes back to being a non-weapon, only capable of healing.
As for advantages, there several. For starters, Sesshomaru is able to slay anything not of this world, for example ghosts and messengers of the underworld. He is able to use this fact to get past Gozu and Mezu, the guardians to the path of the netherworld, since they are not of this world and do not wish to be destroyed by Sesshomaru. He is therefore the only person able to do this. Most other people would have to be slain by Gozu and Mezu or, if they get past them without dying, and open the doors, they would be turned to stone. Sesshomaru is immune to this through the Tenseiga.
By slaying the messengers of the netherworld around a dead body, Sesshomaru can bring anyone back to life once. The only people known to have been revived by Sesshomaru in this way are, in order, Rin (the first), Goshinki's disembodied head, Jaken, Karan, Shuran, Shunran, and Kanta's father. Once someone has been saved, they may not be revived again.
Sesshomaru's father, the Inu no Taisho, also revives his wife (and Inuyasha's mother) Izayoi after she is slain by the human Takemaru.
Additionally, Sesshomaru is theoretically invincible, since the Tenseiga automatically casts a barrier before a fatal blow is delivered and teleports him to safety. This is perhaps the Tenseiga's most practical use.