Information asymmetry is the theory that if one party in a business decision has access to superior data than the other, the decision-making process will be unbalanced.
Signalling theory states that parties in decision-making strive to convey accurate information to each other based on accepted standards.
Pecking order theory states that financial decisions are based on the path of least resistance, using available money first and only issuing equity as a final resort.
To tie these all together, we can first assume that information asymmetry is true, allowing signalling theory to be of importance to all parties involved. Pecking order theory directly utilizes information asymmetry, so the question is whether signalling or pecking order should be followed in making a financial decision.
If we follow signalling, both parties should take steps to ensure that information on both sides is of equal value and importance; one side stands to gain significantly if they signal incorrectly (deliberately or not). This will remove information asymmetry and allow the decision to be based fully on fact.
If we follow pecking order, the decision will be made based on least resistance no matter what the information, and so the first balance of power falls to the business instead of the investor. Only after the business issues equity does the power fall to the investor; while debt is accrued, returns on investment are assumed to be higher than the debt.
The most basic difference is in specifics; signalling refers to any information trade while pecking order refers directly to business investment. However, signalling is an important part of all business transactions.