Does Vesak means Vaishakh? Vesak means on the day Buddha born and is word Vesak (Pali; Sanskrit: Vaiśākha) means Vaishakh month by marathi calendar(Maharshtra, India)?
Vesak is the day marking the birth, enlightenment, and passing away of the Buddha, which according to traditional accounts all occurred on the full-moon day of May. Vesak (by that I mean the three events) has nothing to do with the Vaishakh Month in India.It is just that the three event happen to fall on this month. 

The three great events that took place on Vesak day in the life of the Buddha are - the birth of Prince Sidhartha, the Bodhisatva at the Royal Park of Lumbini, His attainment of the Perfect Enlightement in his 35th year under the shade of Sri Maha Bodhi at Buddha Gaya, and His Parinirvana. His Pari-nibbhana, at Kusinara in the Sala Grove, Upavartana that belonged to Royalty of Malla-occured on the Vesak Full Moon Poya Day. It is renowned as the most sacred Poya Day in the Buddhist world over.

Although the recognised Vesak Full Moon ‘Poya’ (holiday) Day falls in the month of May, there is an additional Vesak Full Moon called "Adhi Vesak Full Moon Poya Day". This Adhi ‘Poya’ occurs once in 3 years, due to the change of planentary movements. There are 365 days in an year. According to astrology (SriLankan ‘Nekath’), they calculate a year usually as 360 days. The planets rotate round the Sun. Sometimes the course of the planetary movements change. According to astrological calculations, this leads to an additional month.

Here I should mention the name of Sir Ponnambalam Ramanthan, who was a Member of the Legislative Council of Ceylon from 1879 to 1892, Solicitor General from 1892 to 1906, a member of the Legislative Council from 1911 to 1921, and from 1924 to 1930. A Barrister by profession, he was elected by a broad coalition of Buddhists and Hindus. He supported the efforts of the Theosophical Society to establish Buddhist denominational schools. He worked tirelessly for Hindu Buddhist unity. He was responsible for making Vesak, the birth anniversary of the Buddha, a public holiday in colonial Ceylon.