What is the difference between drought resistant non-succulent plants and plants that cannot be allowed to dry out?
The difference may be related to how the plants fix carbon. While all plants convert CO₂ and H₂O to glucose and oxygen, there are at least three pathways that are used to do it. The C3 pathway is older and less efficient than the CAM and C4 pathways. Many drought tolerant plants use the CAM or C4 pathways because less water is needed. The C4 pathway is better than C3 in drier climates, hotter climates, and when CO₂ or nitrogen are limited. The hard part about this answer is that I'm not a botanist. Wikipedia lists some plants that use C4, CAM, and C3, but there is no comprehensive list, so I can't match up all the plants you mentioned in your question. In general, a higher percentage of monocots, especially grasses, use C4, but a wider diversity of dicots use C4. But you still can't draw conclusions about families of plants, because while corn and sorghum use C4, rice and barley use C3. Ragweed, a plant you say grows well in dry conditions, is a member of Asteraceae, which includes many plants that use C4. However, rudbeckias are also members of Asteraceae, but don't do as well in dry conditions. The C4 pathway has appeared and disappeared in several different plants, making any general claim hard to make. Here are some links that might help:


http://www.en.wikipedia.org/wiki/C4_carbon_fixation   
http://www.en.wikipedia.org/wiki/C3_carbon_fixation   
http://www.en.wikipedia.org/wiki/Crassulacean_acid_metabolism
http://www.brown.edu/Research/Edwards_Lab/reprints/sage_etal_2011jexb.pdf


If you know more about plants than me, that last link might be helpful, it's about the evolution of C4 pathway in plants and includes lists, but uses the formal species names I just don't recognize.