I naturally was running late and the weather was making my drive slow.
What caused you to have a late start?
I woke up late in the morning and I had a lot of stuff to do.
It's good to be responsible and keep your appointments, as the other person is using up their time, too. If weather is bad, be sure to get an early start so that you can avoid lateness.
I can be late anytime I want.
It's necessary to consider how you impact the schedule of others. If you make plans with someone, it's important to be on time so that you will not be seen as inconsiderate.