Does "help save the planet with your mobile phone" have a different meaning or connotation than "help save the planet using your mobile phone?"
In this case, "using" can be seen as a more explicit description of what to do with the phone. "Using" more or less encases "with" but it doesn't work the other way around:
In the context of a phone and saving the world, the options of doing something with the phone is small and it can be implied that the intention was to use it. Other words have more options:
There are also plenty of other things we can do that completely muck around with the implications:
The context makes the latter two silly but it shows the importance of choosing your words carefully:
This could mean all sorts of things and reminds me of the classic:
Slightly. The first could be interpreted that your phone will be "used up" by the process or used physically (e.g. plugging the gap in the ozone layer by throwing your phone into it).
The second sentence does not have this implication. Ergo, I prefer the second although I'd add a word.
One interpretation could be "have your phone with you while you help save the planet", an implication that the phone itself is incidental to the process. This doesn't make sense, at least not the sense you wish to convey.
Consider: "Use your phone to help save the planet" or similar.
Also, "help save the planet" is a little clumsy as it has two verbs together; consider "help to save" (i.e. use the infinitive) in future instances of this.