Avoid is neutral in meaning; evade implies an element of personal effort often involving cunning or deceit. Typically you evade more seriously unwelcome things such as arrest, detection, identification, taxes, and (quite often) the truth • The drop in revenue was largely due to efforts to evade the poll tax. • For once they managed to evade the searchlights and dodge the guard-dogs. You can avoid them too, but more usually you avoid things that are more routinely unwelcome and more easily dealt with • What can I do to avoid catching head lice? • It is best to avoid continuous hard braking.

Note that tax avoidance is legal, and tax evasion is illegal. In 1979, the UK Institute of Economic Affairs invented the word avoision so as not to have to choose always between avoidance and evasion, but the term has not caught on.