You seem to have many misconceptions about how we see and how that relates to light. I'll address them one sentence at a time:
Why are we able to differentiate between colored objects without the presence of light?
Answer: the premise of the question is wrong: we aren't able to differentiate between colors without the presence of light, because we aren't able to see without the presence of light. Light is what allows us to see. "Seeing" is the act of detecting the light coming from the environment and using it to derive what the environment is like.
Rods help us to see during the dark.
It sounds a bit like you think rods help us to see during the dark... without light ? This is not the case: BOTH rods and cones absorb light and thus allow us to see. Rods help us see in low light because they are more sensitive (to light !) than cones. Nothing can help us see with no light.
We can see colour of the object when it reflects light.
It depends on how much light it reflects ! The correct form of that sentence is "We can see the object when it reflects light". (even more correct would be "when it reflects enough light to trigger the light receptors in our eyes, i.e. rods and cones). Whether we can see its colour depends on whether it reflects enough light to trigger the cones specifically.
But in a dark room(room of any color but the light is switched off and it is night),we still can differentiate between colored objects if we open our eyes for sometime.(although not very much clearly)
Indeed; this is because no room is completely dark. Even more so in our artificially lit world, but even in a moonless cloudy night you may have a few errant photons coming from the stars and Sun (via diffusion in the atmosphere), fireflies, what have you; to be fair in our artificially lit world I haven't experienced it. As far as I know you can get true pitch darkness in things like underground caves (once you're far enough from an exit of course).
Basically, if you can see anything then there is some light. (well, short of your optical cortex confabulating stuff, which in fairness it will do).
How the colors are seen when there is no light?
As I hope you understand now, they aren't.
Do we see as the brain remembers their colour?
I think that may actually be one reason we perceive colour in low-light environments. I'm sure some of it is that there's enough light to trigger the cones a little bit, but I wouldn't underestimate our brain's ability to show us the world as it knows it is as opposed to as whatever information it's receiving right this second says it looks like.
Some references :
The state of seeing things in such low light that we only use the rods is called "scotopic vision" :
Some sources for how "dark" environments still contain low amounts of light (one word for "amount of light" being "illuminance", measured in Lux) :
https://en.wikipedia.org/wiki/Aphotic_zone (thank you to @Johnny for that one)
HOPEFULLY LAST EDIT
On reflection I thought I'd say some more about scotopic vision since it's very relevant to the question. It turns out "scotopic" vision is opposed to "photopic" vision (the vision we have in full lighting, where the cones do the work because the rods are saturated), and in between there is an intermediate range called "mesopic" vision, which is likely the kind of vision we experience when it's dark but we still see color (this is ignoring our brain filling in colors, which could also be a factor and in that case could also happen with scotopic vision - but I didn't find any sources on that).
Mesopic vision is used under lighting conditions that seem to range from low indoor lighting or streetlights to starlight; scotopic vision is used with luminances darker than that. This page is all about it and includes a very nice drawing, sadly it's a pdf so I can't show it in the answer: