To bypass is the synonym for both verbs, although they do not mean the same. To avoid means to prevent something to happen, you keep away, escape from it. For example, misunderstanding, ambiguity, disappointment, injury, mistake and so on can be avoided. On the contrary, to evade something means not to dealing with it. You can evade responsibilities, duties, obligations, problems, issues and so on. Taxes can be avoided and evaded, too. If you avoid taxes, you lower the amount of tax by the means of legal methods. It is also called tax minimization. If you evade taxes, you will get a more serious punishment because it is absolutely illegal. In other words, a person or a corporation intentionally avoids paying tax liabilities. In law, to avoid also means to invalidate. For the verb evade, dodge could be used as a synonym.
