The basic structure of Western music is quite constrained, following certain rules and practices. The Western system of notation, known more formally as staff notation, has been in existance throughout most of Western classical music's history and has its roots in medieval music which had been influenced heavily by Roman culture. Oral traition was originally used; later, in order for music to travel over a long distance a system known as neumes-- small marks above the text of a chant-- was used. Later these were put in different colours and at different heights to form the basis of the staff notation used today, in which symbols represent different note lengths are put at heights to indicate different pitches. Indian music, however, is indigenous to the subcontinent and based almost entirely around oral tradition, especially in the case of folk music where there is no "real" standardised system of education. As such, as mentioned in a previous post, the tradition of Indian music has been based around parampara; the passing down of musical knowledge from a guru to his shisya (students) As such the first system of musical notation was a suprising recent invention, mainly the work of Hindustani music scholar Vishnu Bhatkande.
(Pictures from http://upload.wikimedia.org/wikipedia/en/c/c6/Bhat_notation1.jpg and http://gb.abrsm.org/images/modifiedStaff.jpg.) Bhatkhande's notation for Hindustani Indian music (in this case a tal rhythm) (above) and typical Western staff notation. Both systems use pictoral systems-- Western system using notes, Indian system using characters to represent swara-- and certain symbols to indicate other aspects of music (such as the Western f indicating forte ['loud'] and the Indian cross indicating sam [the starting note of the tal]. However there are quite a few differences. For example, the Western system puts notes at different heights whereas the Indian system leaves swara at the same height.
The fundamental elements which make up the music also differ within the Indian and Western traditions.
Although the Indian concepts of raga and tal are roughly equivalent to the Western ideas of scale and rhythm, the similarities are few. For example, whilst the Western idea of scale is merely a set of pitched notes to be played, the Indian idea of raga is a complex base for improvisation which differs from artist to artist, depending on time of day or year, emotion, and many other factors, made up of (usually) five swara. Similarly, whilst Western rhythm simply describes the way notes are player over time, in Indian music tals are a complex, regulated system of regular swara, played on percussion. The structure of songs is also different; whilst Western music uses verses and choruses, Indian music uses alaps and jors (slow improvised section with percussion) followed by a gat (the only part of the music that is written down) and finally a jhala (extremely fast improvised section with percussion).
And then of course there is the musical culture which exists within India. Whilst Western and Indian musical culture both have some shared themes, such as religion-- Christianity for Western music, Hinduism and Islam for Indian music-- there are notable differences. For example, Western musicians have traditional been taught by their families, nobility or in special music schools, Indian music education is largely based on the interchange of information between teachers and students. Also, in Indian musicians still play a special role in many festivals and rites, particularly in folk communities, this role has been substantially reduced in Western musical culture.
(Image from http://chandrakantha.com/articles/indian_music/folk_media/kamakshi.jpg. An Indian folk musician on a traditional instrument.
The influence of Indian music on Western music has been a recent occurance but is now fairly pervasive. The best known example would be George Harrison of the Beatles, who worked closely with Ravi Shankar (1920-2012), one of the best known Indian sitar players who is often known simply as "Pandit" ("Teacher")
Indian music more generally also influenced the whole psychedelic rock movement in general, which was founded on experimentation and perception-altering states. There was even a sub-genre known as "raga rock" in which musicians deliberately aimed to mimic the styles and instruments of Indian rock. Important bands from this movement included The Yardbirds, The Kinks and to some degree, the Beatles themselves, whose song "Norwegian Wood (This Bird has Flown)" was one of the main impetuses behind the whole movement. Many fusion and progressive rock bands were also influence by Indian music, such as John McLaughlin, a guitarist who formed the 70s band Shakti which fused Western and Indian elements and featured Hariprasad Chaurasia on the bansuri.
In more recent years bhangra has become popular. This is a form of Indian pop music influenced by Western, particularly British rock music which began in the 80s, although its origins stretch back to the 60s when immigrants from all over the Indian subcontinent began to play their instruments in Britain. It has become popular in both Britain and the US; America rapper Jay-Z even remixed one popular bhangra song in 2002. In recent years the success in bhangra has even led to a decline in classical instrument lessons in India as more and more Indian youth seek out Western instruments such as guitar and synthesiser as used in bhangra. Indian pop music has also been introduced to the West through the popularity of Bollywood films and filmi (film music).
(Picture from http://eil.com/images/main/The+Beatles+-+Norwegian+Wood+E.P.+-+Reissue+-+7%22+RECORD-114870.jpg.) The Norwegian Wood EP by the Beatles on which George Harrison plays sitar. It was an important record in raga rock.
Finally, the drone used in Indian music has influenced not only psychedelic rock musicians but also those working in electronic music, particularly ambient. The entire "drone" subgenre of ambient music, developed around the 60s by bands such as the Velvet Underground and still somewhat popular today, is largely based around the Indian classical drone sound.
Modern bands playing Indian-influenced music include the Dhol Foundation, who both teach dhol technique and play bhangra music, founded by Johnny Kalsi, and the Asian Dub Foundation, an openly political band who combine an aggresive punk style and Indian lyrical and musical influences with other forms of music, such as Jamaican reggae and dub. Their song "Real Great Britain" from the 2000 album Community Music can be seen below:
My own knowledge
Dhol Foundation website