For me, it's the difference between a commitment and an estimate.
If I say, I am going to finish my project next week, it is essentially a promise. (Same as I will finish)
If I say, I am planning to finish my project next week, it is more of a statement of how my plans/schedule are at the moment.
Some people will be more against changing their plans than other people. They may consider changing your plans to be going back on your word. If you don't know how the person you are speaking to views planning, you should probably make sure that they know the planned date is just an estimate. In all honesty, if I say I am planning to do something next week, it probably won't happen until the week after, since in my job I often have new, higher priority tasks coming in and will need to adjust schedules.