The rule of detachment is independent of the truth tables. With truth tables you can construct tautologies and with the rule of detachment you can prove interesting things given certain axioms, or statements presumed to be true. A 'proof' is a step by step operation where each step is the assertion of an axiom, a tautology, the rule of detachment or one of various other rules of inference about which we needn't worry. Here's an example. Formal' means involving symbols and rules for their manipulation that can be performed in a mindless way.
Formal manipulations give airtight results and logical conclusions, but are very cumbersome.
As a formal system, the propositional calculus was considered too simple to be interesting to the mathematicians of the late 19th century. In particular, one needs a set of qualities that get at the nature of what a natural number is, and that can be given a formal expression. These qualities are incorporated into formal number theory as axioms, presumed to be true. Of course, if someone suggested 'all numbers are equal', one could incorporate it into a formal system, but we'd recognise the axiom as not describing the behaviour of natural numbers. A set of generally accepted axioms was given by Gieseppe Peano in 1889.