Nikon 28-300mm f/3.5-5.6G ED VR photos are not sharp, why?

That lens (the 28-300) does seem to be notorious for sample variety - i.e. you may get a good one, and you may get a dodgy one. I used to have one, and it wasn't bad, but of course it didn't match up to Nikon's big guns.

On top of that, that lens just isn't going to be as sharp as the 24-70 2.8, which is one of Nikon's top performers. A 10x-ish zoom is more optically compromised than a 3x-ish zoom. Lens design 101.

Unsharp images come down to several factors. If you're shooting at longer focal lengths then you will of course have to shoot at faster shutter speeds to ensure sharpness. Also, stop then lens down a bit and shoot at a high shutter speed, that helps too.

If you want ultimate IQ, then you need to stay with the top zooms and primes. The 28-300 is a fine all in one solution, but you have to take some compromises.

The start of the answer is f2.8 against F5.6! F5.6 is two stops slower than F2.8, and thus needs four times length of shutter opening. Probably the difference between 1/60th and 1/15th of a second at the same ISO setting. Also, the longer the lens, e.g 135mm or 300mm, as opposed to 70mm, the more that camera movement shows up, so you are doubly or triply handicapping your attempts at sharp pics in low light by using a long lens.

You can raise the ISO to get a shorter shutter speed, at the expense of getting more "noise" (graininess). Even so, you are always at the limits of possibility when using long lenses (of lowish max aperture) in poor light. Use a tripod to help, then you only have to worry about subject movement.

You could buy a 70mm to 200mm F2.8 lens to help, or one of the cameras bodies which gives good results when using very high ISO numbers, such as 25000 or above. Both of those options are expensive.

In photography life is a compromise. for example you trade a fast shutter speed at high ISO against image noise.

You have a 28-300mm superzoom for convenience that has to be traded against image quality.

That said..
An f/3.5-5.6 is not the ideal indoor lens. This lens is 1-2 stops slower than your 24-70 and that means slower shutter speed. Longer focal length means that when using longer shutter speeds camera shake is more difficult to control. VR reduces camera shake but doesn't stop your kids shuffling!

How does the lens perform outdoors? If it is soft in bright sunlight you may have a lemon.

One final point. Are you sure it is lack of sharpness and not lack of contrast? I have a Sigma UWA lens that is crisp and sharp and clear in good light but on a dull grey cloudy day I leave it at home because the images look out of focus whereas on close inspection they are sharp but just lack contrast. Superzooms do suffer in the contrast department because of the large number of glass elements and air/glass interfaces sapping the light's energy. Try upping the camera's contrast setting a notch or 2 or adjust it when (if) you process your RAW files.

There are different kinds of blur, each with a different cause.

The fact that you're shooting indoors (likely low light) and are using a lens with average maximum aperture (worse than your f/2.8 lens) suggests you may be experiencing mainly motion blur. The VR system can be used to counteract some of the motion, but it may not be enough depending on how much light is in the scene. (The camera cannot see as well in low light as we humans can.)

As others note, the wide zoom range compromises sharpness. Zooms aren't very sharp to begin with, which is why some photographers use prime (fixed focal length) lenses instead. DxOMark gives it pretty bad marks for sharpness:
http://www.dxomark.com/Lenses/Nikon/AF-S-NIKKOR-28-300mm-f-3.5-5.6-ED-VR

But seeing as the D610 takes fairly high-resolution shots (~24MP), you can regain some apparent sharpness when scaling down the images.

The more zoom a lens has, the lesser optical quality is. Blame science on that.

To get the sharpest photos possible, shoot under great light quality only. Sharpness is more of the photographer's responsibility.

Of course. Cramming that much zoom range into an FX lens is for convenience. It is a heavy sacrifice in quality.
http://slrgear.com/reviews/showproduct.php/product/1360/cat/all