There was no real reason for Darth Vader to go to and look for the remains of one unimportant former jedi. What would she matter to the right hand of the Emperor?
But Anakin. He needed to know. To see for himself if the Togrutan woman he had trained and loved was still alive. If there was one person alive who still knew him as he was. That he loved as Anakin.
When he found Ahsoka's lightsaber in the snow, it was almost like a symbol of all he had lost. The last piece that pushed Anakin further into the darkness. The two most important living women in his life, Ahsoka Tano and Padme Amidala Naberrie Skywalker were dead. And Anakin Skywalker was no more.