Bit depth and dynamic range really are the same thing, provided that
1) System noise is low enough that bits are not wasted digitizing noise
2) The ADC is linear, so each bit represents twice as much signal as the previous bit.
The "steps on a ladder" analogy is flawed, because the ADCs we are talking about are linear. With a linear ADC, extra bits necessarily extend the length of the ladder, as well as providing smaller steps.
Here is a thought experiment to illustrate the connection between bit depth and dynamic range. Suppose we photograph a perfect gray scale, with discrete steps of exactly one f/stop in brightness, using linear ADCs with 8 bits or 12 bits, in a camera with no noise.
The 8-bit ADC will record 8 steps of the gray scale, with output values from maximum white (8-bits = 255) to minimum not-quite-black (1-bit = 1), for a total dynamic range of 8 f/stops. The 12-bit ADC will record 12 steps of the gray scale, with output values from maximum white (12-bits = 4095) to minimum not-quite-black (1-bit = 1), for a total dynamic range of 12 f/stops. Extra bits increase the dynamic range which the ADC can record.
Depending on the relative exposure, the extra steps of the gray scale recorded by the 12-bit ADC can be darker or brighter than the steps recorded by the 8-bit ADC. Let's assume we expose optimally in each case, so that the brightest step of the gray scale which we are interested in recording barely saturates the sensor, turning on all bits in the ADC output. Then the 12-bit ADC will record four steps of the gray scale which are too dark to be recorded by the 8-bit ADC.
With this exposure, each section of the gray scale will be represented with much finer gradation by the 12-bit ADC than by the 8-bit ADC. This is easy to see mathematically -- just divide the output of the 12-bit ADC by 16. Now every discrete step of the gray scale will be numerically the same for both ADCs, but the step size of the 12-bit ADC will be 16 times finer than the step size of the 8-bit ADC.
Another way to think about this is to replace the discrete gray scale steps by a continuous brightness ramp. The 12-bit ADC will record any small section of the continuous scale with 16 times as many discrete output values as the 8-bit ADC. Extra bits provide more resolution across the entire dynamic range.