What is the difference between a reaction of Lewis acids with Lewis bases and a redox reaction?
It's indeed redox - Lewis theory doesn't account for redox, but on grounds of Usanovich's theory ...


A Lewis acid is an electron pair acceptor; a Lewis base is an electron pair donor. The simplest Lewis acid is the hydrogen proton. H$^{+}$ has no electrons by itself. It accepts electron pairs from bases in Lewis acid-base reactions. Here's an example of a Lewis acid-base reaction; note how a pair of electrons from the Lewis base attacks the Lewis acid. 



Redox - a portmanteau of reduction and oxidation - involves the transfer of electrons. Hey, there are electrons being "transferred" above, right? 

At first glance, it seems that all Lewis acid-base reactions would be redox reactions. However, there is an artificial line in the sand that one would do well to heed. 

Redox reactions, as taught at the general chemistry level, involves changes in formal oxidation state. What's an oxidation state? It's the hypothetical charge that an atom would have if all bonding were ionic (as opposed to covalent, or electron-sharing). This means that the more electronegative atom in a bond would have complete possession of the bonding electrons. 

If we go back to the reaction pictured above, then we see that electrons are not being "transferred" in the formal sense. Hydrogen is of much lower electronegativity than oxygen; hence, in a formal sense, it has gained no electrons - remember, when assigning formal oxidation states, we take all bonding as ionic. 

Obviously, in reality, no bond is 100% ionic, but this is exactly the artificial line in the sand I was talking about.


  For example I came across this equation:
  $\ce{2Al_{(s)} + 6HCl_{(aq)}} \longrightarrow \ce{2Al^{3+}_{(aq)} + 6Cl^{-}_{(aq)} + 3H_2_{(g)}}$
  And thought this was a redox reaction but my book says it shows basic nature of aluminium?


This indeed is a redox reaction, as aluminum is oxidized and hydrogen is reduced as we go from left to right. 

It also shows that aluminum can react with hydrochloric acid to make hydrogen gas; this removes hydrogen protons from the solution and thereby makes the system more basic. 

This is not an acid-base reaction in the Lewis sense, as electron pairs aren't being transferred. It's not an acid-base reaction in the Bronsted sense either, as hydrogen protons are not being moved around. 


  On grounds of Usanovich's theory Al as reductor would indeed be basic.


Usanovich defines an acid that accepts negative charge; a base as something that accepts positive charge. Therefore, Al is a base here; it becomes positive from left to right. This theory is an even more generalized view of acids and bases than Lewis acid-base theory.