Yes, there is. First, let me correct your grammar in sentence 2:
We were going to go to the beach, but we changed our minds when it started to rain.
We were going to go to the beach, but we have changed our minds because/since/as it started to rain.
First, you change your mind and I change mine, but we change our minds, since we have more than one. Next, we have changed our minds means that we did so at an unspecified time in the past. When it started to rain is a specification of a time, so you need to use simple past. If you change when to one of the other prepositions I have in the second example, then you are no longer specifying a time at which we changed our minds, so have changed is more correct.
Now, to answer your question: to think better of something is more specific than to change one's mind about something. Thinking better of going to the beach implies that we have decided that it was a bad idea to go to the beach, with the further implication that we decided not to go, i.e. changed our minds about going. Changing our minds about going doesn't have this implication. We might change our minds about going because we decided instead to go to the amusement park, for example.
So, while thinking better of something implies that you also changed your mind, changing your mind about something doesn't mean that you thought better of it.
Here are definitions for the two: change one's mind and think better of. While think better of gives one definition as "reconsider, change one's mind about," you will also see that all of the examples given involve a bad decision that was avoided:
I hope that you will think better of what you are doing and how many people you are hurting.
I will think better of making such a careless remark next time.
I hope you'll think better of it before you quit your job.
I almost bought an expensive watch, but then I thought better of it.
Conversely, examples given for change one's mind don't address the quality of the avoided decision (since there's only one example, I borrowed some from oxforddictionaries.com as well):
I can always change my mind about going on this trip.
He turned to go and then seemed to change his mind.
Before long, I changed my mind and began making plans to go to North Carolina.
However, the manager of that different area changed his mind, so I was in limbo.
Edit: one more example occurs to me, which should explain the difference pretty clearly:
I almost spent the $75,000 I won in Vegas on a watch, but then I thought better of it and paid off my credit cards instead.
I almost spent the $75,000 I won in Vegas on a watch, but then I changed my mind and spent it on a diamond necklace instead.