Yoga has a place in the modern world for several reasons.
The modern world is dealing increasingly with stress. Everything in the philosophy of yoga brings us to stillness.
The philosophy of yoga talks about stopping the fluctuations in the mind. Holding yoga postures for long periods of time bestows silence and stillness on the mind and of course meditation (an important part of yoga practice) brings stillness to the mind. Yoga is very necessary in the modern world.
One reason that historically yoga has been so popular is that it is not just a philosophy that can only be understood and interpreted by scholars or holy men, but it is something that can be put directly into practice. Yoga is and always has been something very practical. Asanas or physical postures can be learnt easily and holding the postures gives one a direct experience of the sense of peace that yoga talks about. It is this direct experience of peace that has historically always made yoga different to other philosophies and it is a primary reason that yoga stands out and has always stood out amongst other philosophies from India.
The very ideal of yoga appeals to most people. Yoga means union and every human being has an innate desire for a feeling of connection. Yoga philosophy also teaches that our original nature is one of peace. Most people spend their lives searching for something that is missing in their lives and they never find it because they are looking outside. Yoga gives people the understanding that really peace is and always has been deep inside them and through meditation, asana practice, pranayama (breath control) and studying of the scriptures, yoga helps people to connect back with their original state of peace. It is this recognition of yoga being a genuine path to finding peace (that is the original sense of peace within us) that appeals to people.
Another big advantage of yoga is that yoga teaches you that you are an individual. Yoga teaches you to work at your own level and your own pace. The benefits of yoga practice, including health and the sensation of peace, are primarily for yourself, so there should not be any competition. The world of competition draws us towards stress and yoga draws us away from this. Yoga teaches you to focus on yourself. And when you do this you are a better person. You cannot have compassion for others until you first have had compassion for yourself. It is this respect for the self, the moving away from the sense of competition and stress and the moving towards a deep sense of peace that makes yoga a special philosophy in modern times.
Overall then, the benefits of yoga practice are very real. In the modern world there are many reasons to feel stress, but yoga was created to give us a very direct and immediate connection with our original sense of peace. It is for this reason that yoga has and always will have an important place in the world.