Very important. Here's short (I hope) story. My mother-in-law has never liked me. Always tension when she would visit and I would find myself trying to please her and trying to say the right things etc to get her to like me - which by the way never worked.
But what I found is that when the in-laws visit was over and they went home, I was filled with depression and absolute dislike for myself because I was not acting like myself - it is an awful feeling. It took many years but finally I gave up the impossible and was just me. She still disliked me but I sure felt better.
So it is important for you own self-worth to be yourself. If you have personality flaws that bother you, then you can change those thing if YOU want to - but be true to yourself.