MacBook – Youtube doesnt play on a specific connection on Macbook Air

I am using a Macbook Air 13" (2013 model).
i5 Intel Processor – 4 GB RAM – 128GB SSD.

A few days ago, when I opened youtube (on Safari) to watch some videos it gave me the error "An error occurred. Please try again later."

I tried later, tried all the steps that Youtube suggests and it didnt work.
Now I cannot play any youtube video on my macbook.
All the other video sites like Dailyotion and Viemo are working fine though.
Using Firefox also didnt work. I have not tried downloading and using Chrome yet.

One day, I took my Macbook to a friend's place. I connected it to her wifi and Youtube was working like before. I came back home and was unable to play youtube videos again. The same error was staring me in the face.

Since then, I have found a proxy 'Proxy Free' to watch youtube. The videos work using the proxy, but the player is not the default youtube player. Also, chosing HTML5 player on youtube didnt work either.

I have then noticed that it works fine on other networks, but mine.
I have a basic modem (it has wifi but I have disabled it) provided by my ISP 'MTNL'. The modem then goes into a Netgear router. PPPoE is on the modem and the router is set to get the IP dynamically.

My wife and brother have Dell Windows 7 laptops and they are able to watch videos on youtube.

MTU is set 1492 on both the modem and the router. I have tried checking my MTU (also pinged it for a fragment test) and the results were fine.

Please help.

EDIT: Today, I clicked on a youtube link on twitter, and the video started playing. Then I clicked on the videos on the right side and they all (3) played too. But then I went to Youtube's homepage, and tried to play a video from there — same error.

I went to twitter and clicked on the same link again — it played.