There's nothing wrong with being sensitive. I'll pick a sensitive person over a heartless one any day.
Whoever told you to "grow up" doesn't know what they are talking about.
Being sensitive depends on a lot of things, including your general mood, emotional state and how triggering something is for you. It will change over time, but there will always be topics or situations that will make you cry more easily than others.
Let me tell you something; When I was a kid it was impossible to make me cry. Friends and family died, and tragedy and pain would constantly plague my childhood, but I didn't shed a single tear. I was raised to keep my emotions buried deep down, and this became problematic. I was a heartless, empty person.
This led to a very violent and unhealthy emotional breakdown in my later years that nearly destroyed me.
Nowadays it doesn't take a lot to make me cry. And I prefer it this way. It's a lot healthier.