For use on a Canon 600D, primarily for travel, landscapes, macro and subjects in motion.
An additional consideration: if I go with the 50-200mm I could also afford a 50mm lens.
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This question is one of the 'Which one is better" types and so most probably would be frowned upon by this community. ;) I had not heard of a 50-200mm lens till I read your question today. I guess you are referring to Sigma 50-200mm f/4-5.6 DC OS HSM Canon mount.
I would only suggest one thing from my personal experience, consider Canon 55-250mm instead of Canon 70-300mm. It is proven to be best of the low price zoom lenses.
I am not aware of anything about 50-200mm other than it is relatively cheap. But looking at your requirement, I think 55-250 is perfect for "travel photography/ landscapes / macro and subjects in motion" because of its sharpness, reach and versatility(to some extent).
Alone, none of them:
Travel photography - it depends where you are traveling to, but typically for single lens solution you want something that can cover both the wide end and the telephoto end (something like the 18-200 or the 18-135, note those are not the highest quality lenses out there)
Landscape - landscape photography is traditionally done with a wide lens, both lenses are too long.
Macro - both lenses don't do macro at all
Subject in motion - this depends too much on the subject, how you want to stop/show motion and lighting conditions, can't really comment on this.
Those lenses can work well in a two lens kit with a wide lens (but not the 50mm, on a crop sensor that's also too long, you need 35mm or wider).
It is not clear from your question exactly which two lenses you are referring to. Canon currently sells or has sold in the recent past: EF 70-300mm f/4-5.6 IS USM, EF 70-300mm f/4-5.6 DO IS USM, EF 70-300mm f/4-5.6L IS USM, EF 55-200mm f/4-5.6 II USM, EF-S 55-250mm f/4-5.6 IS, and EF-S 55-250mm f/4-5.6 IS II. Additionally, Canon has offered several versions of the 75-300mm f/4-5.6 lens. Sigma once made a 50-200mm F4-5.6 DC OS HSM. The Nikon mount is still readily available, but the Canon version is getting a little harder to come by. Some of these lenses are significantly different than others.
Assuming you want to compare the EF 70-300mm f/4-5.6 IS USM to the EF-S 55-250mm f/4-5.6 IS II, for your 600D I would recommend the 55-250. Following is part of my answer to another question that applies here.
The difference between EF and EF-S lenses. AN EF lens must project an image circle large enough to cover a "full frame" sensor or film negative (roughly 36X24MM). An EF-S lens must only project an image circle large enough to cover the Canon APS-C sized sensors that are approximately 22.5X15mm. Since the APS-C image circle is only about 40% the area of an EF image circle, this allows the lens design to be smaller and lighter. Another benefit is that some of the lens elements themselves can be smaller, and thus use smaller amounts of the expensive materials used in modern lens design. In the case of these two lenses, the 55-250 can use smaller amounts of higher quality materials and still be manufactured for a very reasonable cost. The 55-250, while not outstanding, performs much better optically than the 75-300. It is much more comparable to the more expensive EF 70-300mm f/4-5.6 IS USM in terms of optics.
You will get roughly the same optical performance and a more usable focal length range for your 600D in a smaller, lighter lens at a lower price.