Interest: NRE is taxable too
The first surprise is on the NRE side. In India, NRE interest is exempt under Section 10(4), so it feels tax-free. For a US person it is not: the US taxes worldwide income, so both NRO and NRE interest are fully taxable as ordinary income on your US return, and the India exemption has no effect there. NRE interest brings the added sting that India charged nothing, so there is no Indian tax to credit against the US charge.
NRO interest is taxable in both countries. India deducts TDS under Section 195 at about 31% by default, but the treaty caps the tax on interest at 15% if you file a tax residency certificate and Form 10F with the bank. The US then taxes the NRO interest and gives a foreign tax credit for the India tax, up to that 15%. So the practical move is to cap the Indian withholding at 15% so the US credit and the India tax match.
Dividends, the qualified rate, and the credit
Indian dividends are ordinary income for the US, but they can get the lower qualified-dividend rate, the same preferential rate as US dividends, rather than your top ordinary rate. That is because a foreign company's dividend can qualify when the company is eligible for a comprehensive US tax treaty, and India is on the list of qualifying treaty countries. You still have to meet the required holding period, and it does not apply if the payer is a passive foreign investment company, but for ordinary Indian company shares the qualified rate is often available. India taxes the dividend too, and you take a foreign tax credit for the India tax.
Across both interest and dividends, the theme is the same: the US foreign tax credit is limited to the tax the treaty allows India to charge, so anything India over-deducted above that has to be reclaimed from India by filing an Indian return, not credited by the US. Filing the treaty paperwork up front, so the Indian payer withholds at the treaty rate, is what keeps the two sides aligned and avoids money being stranded. A practising CA caps the Indian TDS, reclaims any excess, and gives your US preparer the India-tax-paid detail for the credit.