Information Theory - units of channel capacity
You are talking of capacity of two different types of channel. 

In one case, the channel inputs and outputs are discrete in time.
At the $i$-th time instant, the received signal is $X_i+N_i$ where 
$X_i$ is the received symbol of average energy $E$ and $N_i$ is the 
noise (typically modeled as a sequence of i.i.d. $\mathcal N(0,\sigma^2)$ 
random variables).  The channel capacity of this discrete-time
Gaussian channel $~$is
$$C = \frac{1}{2}\log_2\left ( 1 + \frac{E}{\sigma^2}\right) ~\text{bits per channel use}$$ 
and so 
bits per channel use is the natural metric.  If we are told how far
apart in time the discrete time instants are, e.g. one channel use per
microsecond, then a capacity $C$ bits per channel use can be stated 
as bits per second, e.g. $C$ Mbps for our one microsecond
example.

In the second case, the inputs and outputs are continuous-time signals which occupy bandwidth and so the natural measure is bits per second per Hertz.  There
are more complications involved in making the transition from the continuous-time
channel to the discrete model, and in connecting the bandwidth $W$, the received
signal $P$ and the noise spectral density $N_0$ to $E$ and $\sigma^2$
(see here
for some details), but when all
this is done we get Shannon's celebrated formula
$$C = W\cdot \log_2\left ( 1 + \frac{P}{N_0W}\right) ~ \text{bits per second}$$
for the capacity of the additive white Gaussian noise (AWGN) channel of
bandwidth $W$.  This capacity can also be expressed as $C/W$ bits per
second per Hertz.