Is there a difference between заря and рассвет? My understanding is that заря can mean both dawn and dusk, while рассвет can only mean dawn, but I'm not sure if this is correct. [Edit: it's not correct, see answers.]
Заря means "twilight", "illumination of the Earth's lower atmosphere when the Sun itself is not directly visible because it is below the horizon" (from Wikipedia).
As with the English word, it can be applied both to light seen before the sunrise and that seen after the sunset, but the morning meaning is the primary one. If you are speaking of the time after the sunset, you should specify вечерняя заря.
There is another word, сумерки, which technically means the same, but its evening meaning is the primary one, as with the English word "twilight". This one is usually used to translate "twilight".
Рассвет is the brightest part of the morning twilight, seen right before the sunrise. It's the direct translation of dawn indeed.
Both рассвет and заря can be used metaphorically, meaning the time when such a light can be seen (встал на заре, проснулся на рассвете), and, further extending the metaphor, meaning "beginning of something" (заря новой жизни).
"заря" and "сумерки" do not "mean the same", like the accepted answer incorrectly suggests.
The former, is the red coloring of the sky before sunrise or after sunset. The latter is a specific time of day (there are actually notions of "civil twilight" - "гражданские сумерки", "nautical twilight", etc.), and/or a reference to a specific lighting conditions, darker than the daylight, but lighter than night.
These two words do not mean the same thing at all. In fact, the only thing that relates them is that the former usually (not always) happens during the latter.
I think, the problem here is that "заря" has two different meanings. The first meaning, one being mostly discussed here, is, like I said, the beautiful red coloring of the sky that can sometimes be seen just before the sunrise, or soon after sunset. I don't think there is actually a specific English word for this, other than "red skies". https://en.wikipedia.org/wiki/Red_sky_at_morning
A closely related Russian word (the word is related, not the phenomenon) is "зарница", which corresponds to English "heat lightning".
The other meaning is indeed synonymous with "рассвет" (dawn), signifying the beginning of day (or, more widely, the beginning of a new period): "на заре ты ее не буди ...", "заря новой жизни..." etc. In such context, the proper translation would be "dawn".