Cameron and Sturgeon met in Edinburgh for their first talks since Britain's national election last week, which saw Sturgeon's pro-independence Scottish National Party gain unprecedented national influence.
Speaking after the meeting, Sturgeon said the two agreed that previously agreed plans to transfer greater governing powers from London to Scotland should be implemented as soon as possible. But she said she and Cameron disagreed on whether Scotland should control all of its tax and spending policies.
Cameron has pledged to introduce legal changes later this month that would allow the Scottish government to control policies including the rates of income tax and some welfare spending.
That does not go far enough for Sturgeon, who says Scots want to take more substantial control over their economy, including the right to decide their own minimum wage, employment law and business taxes.
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The prime minister said he remained committed to delivering a stronger Scottish parliament, and would consider "sensible suggestions." But he also pledged to keep the four parts that make up the UK, Scotland, England, Wales and Northern Ireland in one united nation.
"I'm a great believer that in our United Kingdom we want a strong Scottish parliament, but we also want the solidarity between the different parts of the UK and I don't want to lose that," he said.
"He can act as if it is business as usual and nothing has changed, and people will draw their own conclusions from that," she said. "They will think that Westminster isn't capable of listening or responding.