It’s not of the Sunnah to soak up the remaining drops of water after wudu’. However, one can do so for valid reasons. Dr. `Abdel-Fattaah Idrees, professor of Comparative Jurisprudence at Al-Azhar University, states: “Using a towel after performing wudu’ for the purpose of absorbing the remaining drops of water on one’s skin is contrary to the Sunnah of the Prophet (peace and blessings be upon him).
However, if one fears that his body will be at stake as a result of pollution from the specks of dust flying in the air at the place of performing wudu’ or one finds it necessary to remove the drops of water due to extremely cold weather, then there is nothing wrong in using a towel to soak up the remaining drops of water.
On the other hand, if the weather is relatively mild or even hot, then one should refrain from using a towel or any other means to remove such remaining drops.
Really, soaking up the remaining drops of water following wudu’ is an act not recommended in the Shari`ah, for in doing so, one removes the traces of an act of worship (wudu’), which is originally ordained by Allah. Hence, one should refrain from such act unless it really causes harm.”
In conclusion, we would like to inform our dear brother that the hadith he referred to in his question is an authentic hadith, which reads:
Maymunah (may Allah be pleased with her) said: “I placed for the Prophet (peace and blessings be upon him) water for the bath, and he poured water with his right hand on his left and washed them. Then he washed his private parts and rubbed his hands on the ground, washed them with water, rinsed his mouth and washed his nose by putting water in it and blowing it out, washed his face and poured water on his head. He withdrew from that place and washed his feet. A piece of cloth (towel) was given to him but he did not use it.” (Reported by Al-Bukhari)