It's the same as in humans in that giving the drug by the intravenous route means that it can get to work straight away. However, sticking a needle into a vein is invasive and can cause infections, vein damage etc.
When given orally, the drug first has to dissolve and then negotiate the GI tract and be absorbed into the blood stream. This takes time and the amount of drug available may depend on various factors eg absence/presence of food, whether there are any obstructions which may impede the absorption process, whether the animal is able to take the medication etc.
Financially, intravenous injections will be more expensive and you will (probably) need a vet or nurse to give it, adding to the expense.
In most cases, one would chose to use the oral route where possible, reserving the intravenous route for emergencies, initial dosing to get the levels up or where the drug isn't available in an oral formulation.
In the case of severe adverse side effects eg anaphylaxis, you would expect them to normally occur on the second or subsequent dosing - if they were to occur when the injection was being given, they would be fairly quick to show and the vet would be able to give things like hydrocortisone to counteract them.