I do not know of any books that make such comparisons their main theme, but
texts on the categorical approach to universal algebra will often also discuss
how their approach relates to "traditional" universal algebra.
I think that Lawvere's Thesis from 1963, available as reprint with commentary
is probably the best way to start. Beside the obvious advantage of being freely
available, it is from the main inventor of this type of categorical algebra,
includes comments from 2004 on subsequent developement and also has additional
references. Textbook treatments are the article of Pedicchio and Rovatti, the
book of Pareigis and the small book of Wraith (all cited in the references of
the above on page 20f).
Another good textbook treatment along with discussion is provided in Chapter 3
of the book by Borceux "Categorical Algebra II".
Francis Borceux, Handbook of categorical algebra. 2,
Encyclopedia of Mathematics and its Applications 51,
Cambridge University Press, 1994
Slightly older textbooks are e.g.
Ernest G. Manes, Algebraic theories,
Graduate Texts in Mathematics, No. 26,
Springer-Verlag, New York 1976
Günther Richter, Kategorielle Algebra,
Studien zur Algebra und ihre Anwendungen 3,
Akademie-Verlag, Berlin 1979