The saving clause is the provision every US citizen abroad has to understand, because it undoes most of the treaty for them. Nearly every US tax treaty contains a clause under which the United States reserves the right to tax its own citizens and residents as if the treaty had not entered into force. The practical effect is that a US citizen generally cannot use the treaty to reduce US tax on most of their income, even income the treaty appears to assign to the other country. The saving clause carries a short list of exceptions, typically covering items like certain pensions, child-support and social-security provisions, and relief from double taxation, but the default is that the treaty works for the foreigner in the United States far more than for the American abroad. This is why the foreign tax credit, not the treaty, is usually the tool that actually prevents double taxation for a US citizen.
Reduced withholding is the benefit treaties deliver most reliably. When one country's resident receives dividends, interest, or royalties from the other, the source country would ordinarily withhold tax at its statutory rate, which for US-source dividends paid to a foreign person is 30 percent. A treaty commonly reduces that to 15 percent or less on dividends and often to zero on interest and royalties, provided the recipient qualifies and, in the United States, generally documents the claim on a Form W-8. This is where a treaty most directly saves money, and it flows in both directions.
The residency tie-breaker resolves a genuine conflict. Two countries can each conclude, under their own domestic rules, that the same person is a resident, which without a treaty would expose the person's worldwide income to both. Treaties supply a cascading test, usually permanent home, then center of vital interests, then habitual abode, then nationality, and finally mutual agreement between the tax authorities, to assign residency to one country for treaty purposes. For a US citizen the tie-breaker's value is again limited by the saving clause, but for many resident aliens it is decisive.
Claiming a treaty position that overrides the code has to be disclosed. When a taxpayer relies on a treaty to take a position contrary to US tax law, for example to exempt income the code would otherwise tax, the position generally must be disclosed to the Internal Revenue Service on Form 8833, the treaty-based return position disclosure. Failing to disclose a position that requires it carries a statutory penalty. Not every treaty benefit needs a Form 8833, and there are exceptions for routine items such as reduced withholding on certain investment income, but a substantive override of the code generally does.