To me, Christmas means the birth of Jesus Christ (because I'm Catholic) but I won't go to much into detail into that since people believe in different things. Christmas is also the holiday of giving and receiving, not only presents but giving and receiving love from people you know most of the time. That doesn't mean you can't be kind to strangers too, especially to the ones that don't have much.
To me, New Years is celebrating the beginning of the new year and reflecting on what we have accomplished during the past year(s). It's also my sister's birthday and my brother's birthday two days later so we usually go to somewhere extra special.