Answer: If both were struggling, we'd consider the type of care, but since one of them is doing well, it is likely that the second is just not as healthy and vigorous a plant. This past winter's freeze was hard on many plants. Since it sounds like one of the honeysuckle bushes was already struggling, it is likely that the frost struck it harder, and it has not been able to bounce back quickly. There's still time for it to recover though. I wouldn't pull it out just yet.
Here are some other possibilities to consider when two of the same species planted in the same area and supposedly receiving the same care respond so differently. One may have good drainage and the other have poor drainage. One could be getting more water than the other. The soil could be compacted where one is located. There could be construction debris under one, inhibiting root growth. One could have been root bound when planted so the roots continue wrapping around themselves, inhibiting growth. Has there been any construction activity near the plant? Heavy equipment compacts the soil and can damage roots. Have you applied any weed killer near the tree? Any chemical drift from the neighborhood?
Finally, make sure the water is penetrating through the plant's entire root zone, at least 1 foot deep, maybe 2, depending on the size of the plants. Use a soil probe (any long, pointed piece of metal or wood to poke into the soil) to check how far water has penetrated. The probe moves easily through moist soil, but stops when it hits hard dry soil. Good luck!
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