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First of all, the Melodyne studio bundle contains two products: Melodyne studio and Melodyne editor. Melodyne editor comes with the advanced DNA technology to edit polyphonic audio, Melodyne studio doesn't. Melodyne editor can run as a plug-in in a compatible DAW, whereas Melodye studio runs as a program on its own while utilizing the MelodyneBridge to connect to a DAW. Melodyne studio features multi-track arrangements (including quantization to reference tracks, for example), while Melodyne editor is single-track. Melodyne studio has a few more MIDI options than Melodyne editor.
See also our product comparison chart
Both programs should run fine with the latest version of Logic Studio.
If this doesn't help, please let us know what exactly you want to do with Melodyne.