By Paul Reynolds BBC analyst

It was striking during this visit that President George W Bush was received with greater enthusiasm in Russia than he was in Germany or France.

He cemented his new friendship with Russian President Vladimir Putin with an agreement on reducing the number of deployed nuclear weapons and by backing a new Nato-Russia Council.

The visit has raised the fundamental issue of where transatlantic relations are going

But he could only paper over the cracks which have opened up between the United States and some of its European allies.

To some in Europe, George W Bush is still the Texan gunslinger, whose ignorance is no longer a laugh, whose "war on terrorism" (with perhaps a war against Iraq in the planning) is dangerous, and whose understanding of the Middle East is shallow.

To Mr Bush and his administration, that kind of talk is European whingeing at its worst.

The same whinge, they might say, was heard when Mr Bush proposed pulling out of the Anti-Ballistic Missile Treaty.

All manner of mayhem was predicted, yet that issue was hardly mentioned on this trip. So much for European hand-wringing.

Thin glitter

Not that George W accepts that his visit emphasised differences. He talked about the positive, as all leaders do on official visits.

Bush faced protesters in Germany

He even challenged one American reporter to justify a question about protests in Europe, saying that protestors were good and democratic.

He scoffed at the reporter's question to the French President Jacques Chirac, in French: "Very good. The guy memorises four words and he plays like he's intercontinental."

But even if you put aside arguments as to where he was most or least welcomed, the visit has raised the fundamental issue of where transatlantic relations are going.

There was glitter on the surface but rust is showing underneath.

Moral claims

The fact is that the United States is currently so powerful that Europe is lacklustre in contrast. It still reflects Henry Kissinger's lament that it does not have a "single phone number".

The difference in approach is very marked. Whereas Mr Bush, supported by a majority of Americans, sees the strong US role in conducting a broad-based "war on terrorism" as essential for freedom in the world, his European critics see the same policies as aggressive and unilateralist.

Mr Bush shouts "Wake up, Europe". Europe replies "Listen up, America."

Both sides claim the moral high ground - the Americans as defenders of freedom, the critical Europeans as exponents of a new morality.

One side has chosen force as a way, as it sees it, of expanding security; the other has turned against force because it feels that violence leads to violence.

Europe (or an element of it) is resisting the Bush call that "if you are not with us, you are against us".

That is the underlying tension across the Atlantic. And one visit cannot change it.

Tough policies

But don't forget that transatlantic tension is not new.

Back in the 1980's, there were huge rows over the deployment of American nuclear tipped Cruise and Pershing missiles.

All that has gone. The argument now is whether old American bases in Britain should be used for housing.

Washington - and its supporters in Europe - argue that this shows that tough policies are correct if pursued in the right cause.

The Soviet Union, they say, collapsed and now Russia is giving in, too. George Bush follows that line.

And if critics in Europe don't like it, his message is, he will listen to them but will bypass them.

The scene is set for interesting discussions about Iraq. They could make the suppressed disagreements on this visit look very mild indeed.