They're very different words. 'Plagiarise' is similar to poach (copy without permission and pass off as your own) whereas 'emulate' means to equal, in this case his speech was as good as yours.
To me it sounds odd to say he emulated my speech. Generally you emulate an achievement.
They do not mean the same. Plagiarism is always considered dishonest and often criminal. Emulation is usually seen as complimentary ["Imitation is the sincerest form of flattery"]. "Poaching" usually refers to hunting without permission, but by extension means "stealing [someone's work, etc.]."