Schmaltzy stuff aside and looking at this in a more practical way. Making friends on the road means that you've got someone to watch your bag while you use the toilet. You have someone to share the costs of accommodation, transport and food with. You have some one to beat at cards, drink local moonshine and share stories with; as well as someone to revel in the peculiarities and extraordinary moments you are bound to encounter as you both travel from place to place. Don't get me wrong, with a positive open minded approach you'll have an amazing time on your travels if you do it solo. But, with a friend you've got someone with which to share those special moments as well as those, lets say, hairy times we all tend to encounter on our journeys.
I've made friends in the hostel, on the free tour and from a cafe that for a number of reasons have simply been companions for the day. The random encounter I had on a boat in Bangkok with a lovely Japanese girl, led to her and an amusing French guy (whom I have no recollection of ever being introduced to) both joining me on my explorations of the many bead and plastic carrier bag shops of the China Town area of the city. Bizarre, but great fun; especially when we broke bread (well noodles) together at one of Bangkok's famous street stalls! Just one look at the 20p toothpaste squeezer I bought that day instantly brings back floods of memories and laughter from four years ago.
So, my message to you, is embrace the people you meet on your journeys. Some may become firm friends, others friends for a day, but, all will provide you with a host of memories and experiences that will almost certainly make your travels much more enjoyable.