What's the difference between pink tea and other types of tea?
It's hard to know exactly what you're referring to without any context of the event, location, or what you remember it tasting like, so I'll give you my best educated guesses.


If your pink tea sweet and fruity or floral, it was an herbal tea - tisane, and not true tea. ("Tea" actually refers to the plant whose leaves are used the beverage. Anything that is not made from the tea tree is usually called a tisane.) Some varieties of tisanes that produce a pink liquid include hibiscus, rose petals, rose hips, strawberry.
If the above is not the case, I suspect you're referring to Kashmiri pink tea, sometimes called noon chai.


Disclaimer: I can't find any authoritative sources about Kashmiri online; my knowledge of the subject is anecdotal from my own travels in India as well as stories from people I know who have lived or traveled in South Asia. My understanding is that, like masala chai throughout South Asia, pink tea has some general guidelines but everyone customizes it according to taste or local custom.

The basic distinguishing factors of Kashmiri pink tea compared to a "regular" black, green, oolong, or white tea are twofold: color and taste. As you mentioned, this tea is pink in color, which other beverages made from the tea tree are not. The pink color comes from the addition of baking soda. As for taste, pink tea tends to elicit mixed opinions about taste because it is salty; most people around the world typically drink tea unadulterated, or sweetened, but rarely with the addition of salt.

Pink tea shares many ingredients with masala chai, often including some or all of the following:


Ginger
Cardamom 
Cinnamon 
Peppercorns
Milk or cream 
Sugar (or other sweetener)


Unlike masala chai, which is normally prepared with black tea, Kashmiri pink tea uses green or oolong tea leaves.

Ingredients included in Kashmiri chai that are not included in most other chai recipes:


baking soda (just a pinch)
ground pistachios and/or almonds
salt
white poppy seeds


The final major difference is in the method of preparation. Masala chai can be made in a matter of minutes by boiling the water, then adding tea, milk, spices and boiling it all together for a few minutes before straining and serving. Kashmiri chai, on the other hand, takes 1-2 hours to prepare (I believe the long boiling time is required for the baking soda to accomplish its task of turning the tea pink). I don't know the exact process and couldn't find any sources that seemed reliable.