The 35mm equivalence is the same for both cameras, 450mm in this case. Both are 1.5x DX cameras.

Of the two lenses, the 18-200mm in general seems to be the sharper lens. Whether or not you can see the difference in your actual photography would depend on multiple factors.

The big difference here is the doubling of the resolution of the cameras. The 24mp D5200 is going to show any bad habits in poor holding technique that the 12mp D5000 let you get away with.

All that said, some example photos with full EXIF data are always helpful for diagnosis.

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Joseph K
Seattle, WA, USA

D700, D200, D70S, 24-70mm f/2.8, VR 70-200mm f/2.8 II,
50mm f/1.4 D, 70-300mm f/4.5-5.6 VR, 18-70mm f/3.5-4.5 DX