Treaties stop you from paying tax twice on the same income

One of the purposes of treaties is to ensure that you do not end up in a situation where you are taxed in both countries.  They do this by the countries negotiating which one gets to keep your tax dollars.

Two important things to remember:

One, most treaties do not relieve withholding tax (that tax you have to pay before you file a tax return to calculate the actual amount owing).  But most countries have a mechanism to reduce that amount of withholding if you are not going to owe tax in that country.

Two, if you have paid tax that under the treaty you should not have paid you cannot claim a foreign tax credit for that amount.  You need to go back to the first country and request a refund of overpayment.