The question pretty much explains itself. How do the two methods compare? I've always used Assembly PCR but the method is prone to mistakes and I'm curious how it compares to Ligase Chain Reaction (LCR).
I have used both Gibson assembly (what you mean by assembly?) and LCR as replacements for blunt or sticky ended cloning. LCR has, for me, worked better in that a higher proportion of constructs that come out have been assembled correctly and I have noticed no mutations after LCR while I have seen them somewhat often with Gibson.
In any PCR reaction ( including the assembly PCR ) the substrate of the reaction is a mix of mono nucleotides. Specific oligos serve as primer by annealing on the template and that are extended by a DNA polymerase. It is a polymerization reaction. In the LCR there are no mono nucleotides instead you have many different oligos that anneal one after the other on the template and a DNA ligase is used to join them together to form a longer DNA molecule. It is a ligation reaction. This two reactions have much in common. Firstly, they rely on the annealing of oligos on a template. Secondly, both the polymerase and the ligase need to be thermostable. In fact, both reactions leverage the thermostability of the enzymes to carry on cycles of annealing, reaction (polymerization or ligation) and melting. After each cycle the template DNA is copied so you end up with an exponential amplification of your template with both techniques.
Here a nice presentation that showing these concepts with some graphics.