ND filters

Advantages

• No extra post-processing required.
• You can see the result in the viewfinder.

Disadvantages

• Making the exposure is more complicated because you have to select a filter and place the transition appropriately for the scene.
• You probably need several filters (of different density and transition abruptness) to cover a sufficiently wide variety of scenes.
• The straight transition between dark and light needs to more or less match the needs of the scene (for example, a large mountain with sunlit clouds behind probably won't work).
• ND filters are more fragile than other filters because they are placed in a frame so you can shift around the transition.
• Good ND filters aren't cheap, and you may need multiple sets to fit different lenses.

HDR

Advantages

• No mucking around at exposure time (just make the exposures).
• Can deal with arbitrarily shaped transition between light and dark areas.
• A variety of HDR software is available for free (though there are pay options as well).

Disadvantages

• Lots of mucking around in post.
• If your camera doesn't support bracketing with a sufficient number of frames, a tripod will be required and you'll have to adjust exposure manually.
• Scenes with moving objects will lead to difficult or impossible editing in post.
• Significant practice is required for good results, particularly if you don't want the "HDR look".