It's very hard to be sure about the "constant white noise" on the recording. It's not something that troubles me. As the linked post says, I used a Rode "dead kitten" to suppress wind noise though the wind was quite light. But there were plenty of trees and bushes around, and if you think of those thousands of leaves rustling even a little, that makes a certain amount of whitish noise for a start. I think our ears naturally exclude some general environmental noise when we listen naturally, and it only becomes obvious when listening to a recording.
The other thing to consider is playback level. For me, a recording like that should be played back at a level such that (for example) the footsteps on the gravel sound no louder than footsteps on gravel should sound (my feet, not Shrek's!). The whole recording should be heard at that correct level, so your aural experience is authentic. Of course you can crank up the quiet bits and hear system noise, but then the playback ceases to be authentic.
Without having carefully tested, I don't think there's a significant difference in the noise from the built in mics of the Tascam and Zoom recorders, but part of the shootout I will conduct later will be to provide some kind of noise floor comparison via a ticking clock test. That will be part 2 - part 1 will simply be a recording of my studio's monitors replaying one minute of a Jacques Loussier track, with careful attention to the placement of the devices in the same spot and with level setting being the same. It won't sound particularly good (recording loudspeakers usually doesn't, but of course recording a rock band's PA speakers isn't really much different) but having the very high quality Sennheiser mics as part of the test will provide a basis for comparison. The test should also reveal differences in stereo imaging, which for me is important as well as frequency response.
As for robustness - well, you can get a hardish case shell for the Zoom, but I don't think one is available for the Tascam. The Zoom perhaps has more vulnerable switches, but I don't think there's a significant difference between the robustness of the two. If you can, treat any such device as gently as possible. On your trip of course that may not always be possible!