What is the difference between serial correlation and having a unit root? I may be mixing up my time series and non time series concepts, but what is the difference between a regression model that exhibits serial correlation and a model that exhibits a unit root?  
In addition, why is it that you can use a Durbin-Watson test to test for serial correlation, but must use a Dickey-Fuller test for unit roots?  (My textbook says this is because the Durbun Watson Test cannot be used in models that include lags in the independent variables.)
 A: A simpler explanation can be this: if you have an AR(1) process
$$y_t = \rho y_{t-1} + \epsilon_t,$$
where $\epsilon_t$ is white noise, then testing for autocorrelation is $H_{0;\mbox{AC}}: \rho=0$ (and you can run OLS which behaves properly under the null), while testing for the unit root is $H_{0;\mbox{UR}}: \rho=1$. Now, with the unit root, the process is non-stationary under the null, and OLS utterly fails, so you have to go into the Dickey-Fuller trickery of taking the differences and such.
A: If you have, say, an autoregressive process, and you look at what is called the characteristic polynomial, that polynomial has complex roots (maybe some or all are real roots). If all the roots are inside the unit circle the process is stationary otherwise it is non-stationary.  A test for unit roots is looking to see if the specific process is stationary based on the observed data (parameters unknown).  
A test for serial correlation is entirely different.  It looks at the autocorrelation function, testing to see whether or not all correlations are zero (sometimes referred to as a test for white noise).
The answer to the second question is that different problems require different tests. I don't understand what your book is describing.  I see these tests as tests on individual time series.  I don't see where independent and dependent variables enter into it.
