A cultural system can have absolutely opposite connotations in different social and religious context. When we look at Hinduism and Islam, dance and music have exactly opposite connotations in these two religions, but there have also been efforts in history to bridge this chasm and merge the two traditions through dance and music.
In Hinduism, all forms of artistic activity - whether it is painting, sculpture-making, architeture, dance, music, singing or any other artistic activity - all of it is essentially a sacred act. The theme of the artistic activity need not be religious; just the activity of indulging in any art form is a sacred act. Indeed, there is Saraswati the patron deity of arts, learning and knowledge, there is Shiva, the master of dance - who creates, destroys and sustains the universe through his dance, there is Krishna, who is an expert dancer, and when the Goddess dances, the whole universe dances with her. There are many other deities who take the role of a dancer, musician or a singer from time to time. Add to this the multitude of sages who are forever singing the praise of the divine.
Because of this view of arts, Hinduism in general is flooded with artistic activity, art forms were patronised by the rulers and artists were attached to the temples, granted with money and land and performed their artistic activity in the temple precincts on festival days. It has always been considered an accomplishment in Hinduism to be an exponent of any art form.
Because of its sacred associations, art activity in pre-modern India was almost exclusively religious in nature. Even today, the folk arts and the classical music and dance traditions are predominantly religious in character, even though they have been brought out of the sacred space and are performed in a secular environment for a secular audience for commercial purpose, but the theme is almost always religious. It is impossible to understand the traditional art forms of India without understanding the Indian religions.
Here, I'll only talk about Odissi, a classical dance form from Orissa the Eastern state of India in brief.
Odissi, like all other classical dance form of India, was a temple dance, with dancers attached to the huge temples of Orissa such as those at Puri, Bhubaneshwar and Konark. These temples patronised dance and music traditions and received large grants from the kings for this purpose. However, this centuries old dance form by early 19th century had begun to get degraded into a kind of prostitution as with the growth of the colonial powers in India, royal patronage to the temples were cut off, the British were not patronising this system and the temples lost a lot of their revenue. This was coupled by the excessive priestly orthodoxy and the control of the elites over the society. As a result, dancers began to be exploited by the priests and the traditional royal families - much diminished in political power, but still holding a ritual validity (even today, before the chariot festival of the Jagannath Puri, it's the earlier royal family's descendant who comes to perform the "royal rituals," not the chief minister who really holds the political power). Hence, by the late 19th century the system of temple dancers (called Devadasis) attached to the temples began to diminish and they were now working as dancers as well as temple prostitutes. With this, the classical dance tradition of Odissi began to die, as was also the case with other classical dances of India.
After Independence, the Indian government decided to promote the various art forms of India which were on the verge of extinction by this time. In every region, some exponents of art forms came to recover the knowledge from the ancient texts, from the few surviving artists and connoisseurs. In Orissa, Kelucharan Mahapatra collected the dance postures as described in the ancient dance treatises, as practised by the folk dance form called Gotipua in which young boys dress up as girls and dance on the song and music related to Krishna's stories, from the few exponents of Odissi left in the temples and from the iconography of dance sculptures from various temples of Orissa. In 12th century, there was a great Sanskrit poet called Jaydev in Orissa who composed a work of poems on Krishna in simple Sanskrit, called Gita Govinda (see the multimedia project here). This text was made the primary basis for enacting the classical Odissi. he pieced together all this information and evolved an elaborate system of classical Odissi which could be performed on stage in a secular environment for a secular audience. He began to perform this Odissi and also taught the women who wished to learn it from him. Thus, the credit for the Odissi as it exists today goes to Kelucharan Mahapatra.
The first Odissi video given above is an enactment of a Sanskrit poetry in praise of Vishnu, one of the three major Gods in Hinduism. The dancer is Sharmila Mukherji, a disciple of Kelucharan Mahapatra. The second video is an enactment of the ten incarnations of Vishnu (see the first comment below for more descriptions). The dancer is Saumya at Bangalore. The third video is about Gotipua, a folk form from Orissa from which modern Odissi has borrowed some postures.
Praise to the Ten Incarnations of Vishnu in Odissi by Saumya
In contrast to this, in orthodox Islam, all forms of artistic activity are prohibited as "anti-religious." Especially, dance and music are forbidden except the drum-music. However, there is a controversy about it, since there is a Hadith in which Prophet Muhammad said to Aisha who prepared a bride for marriage and took her to the venue of marriage, that she should have arranged for (dance and?) music because the Ansars, the community in which marriage was being held loved dance and music. Because of this, many Muslims enjoy dance and music, while the militant Wahhabis and their successors forbid it and also don't allow others from engaging in it. In countries such as Saudi Arabia, dance and music performances are forbidden, with strict punishments for those found indulging in it.
In South Asia, the Muslim community always took a liberal view towards this issue. Especially the Sufis have contributed greatly for melodious music, singing and dance which employs fast whirls. These performances were - and are - held in the Dargahs of Sufi saints and later in the 20th century were brought out of the Dargahs and began to be performed on stage in a secular environment. People from all communities enjoy Sufi music in the Indian sub-continent. In recent decades, pop music singers began to use the rhythms and the beats of Sufi music for their compositions. Runa Laila, a pop music singer from Bangladesh was amongst the earliest to do this. Her composition "Damadum Mast Kalandar" which actually uses a Sufi song, became an all time favourite. The last video below shows a performance of this song by Runa Laila.
The Mughals, as in all other cultural and political fields, take a prominent place in this field as well. During the Mughal rule, all the classical dance and music forms of India were greatly patronised by the Mughal Emperors, under whose patronage some of the musical instruments and musical modes were also invented. Kathak, the classical dance of North India, received extensive patronage during this period and hence, it spread over half of India. To its already existing repertoire from the Hindu tradition, it added the Persian dance elements, since Persian dancers were always coming to the Mughal court. Especially in the later Mughal period, when they had become politically weak and the British were really ruling India, the Mughal princes contributed greatly towards the development of Kathak. Wajid Ali Shah, the ruler of Oudh (approximately Uttar Pradesh and Uttaranchal of today in North India) was a Kathak dancer himself and used to hold Kathak performances in his court. he had received training in classical singing and composed poetry for his Kathak performances. Eventhough he was not successful politically and was eventually captured by the British and sent into exile, he contributed greatly to the dance and music of India.