* Georgia and Russia make some progress in talks

* Two sides to meet again July 1

(Adds quotes from U.S., Russia, Georgia)

By Stephanie Nebehay and Jonathan Lynn

GENEVA, May 19 (Reuters) - Georgia and Russia made some progress on security arrangements in the Caucasus on Tuesday after Moscow's negotiators returned to the table, but major differences remain, international mediators said.

All sides agreed to meet again on July 1 for talks intended to head off any further conflict in an area seen by the West as a key transit territory for Caspian gas and oil and by Russia as a historic sphere of influence. "The process is back on the right track after some problems. The process is in full swing," Pierre Morel, special representative of the European Union for the Georgia crisis and one of the co-chairs of the talks, told a news conference.

But U.S. envoy Matthew Bryza said it was not clear that the Moscow-backed delegations from Abkhazia and South Ossetia -- the two breakaway regions at the heart of the dispute between Moscow and Tbilisi -- had agreed to back the incident response mechanism designed to prevent flare-ups turning into conflict.

Bryza, deputy assistant Secretary of State for European and Eurasian Affairs, said human rights were weakly protected in the two regions and said he hoped the next round of talks would tackle the problem of getting aid to people across the ceasefire lines separating the two regions from the rest of Georgia.

"There's nothing we do that's more important than making sure that every needy person regardless of ethnicity is getting the assistance they need. So we have more work to do," he said.

QUESTIONS OF STATUS

The talks were launched after Russia and Georgia fought a brief war last August over South Ossetia.

The international mediators of the talks -- the EU, the United Nations and the Organisation for Security and Co-operation in Europe (OSCE) designed the talks to focus on practical security and humanitarian problems and avoid the question of South Ossetia and Abkhazia's political status.

But that question nearly wrecked the latest round of talks when Russia and South Ossetia walked out after Abkhazia refused to take part because of a delay by the U.N. in producing a report on its mission in Abkhazia.

Abkhazia had been concerned that the report would imply it was part of Georgia. The report finally appeared on the night between Monday and Tuesday, with the official title of "United Nations Observer Mission in Georgia" but skating round the question of whether Abkhazia is part of Georgia or not.

The appearance and wording of the report allowed Russia and the two regions to join the talks. Both Russia and Georgia complained the report was unbalanced but acceptable.

In the report, U.N. Secretary-General Ban Ki-moon said tensions between Georgia and Russia were weighing heavily on the region.

Russian Vice-Foreign Minister Grigory Karasin said Russia would shortly start to negotiate a Security Council resolution in New York on a new mandate for the U.N. mission in Abkhazia and hoped negotiations would resume in Vienna on the OSCE peace monitors mission in Georgia and South Ossetia.

He said the atmosphere of the latest talks was "tough".

"That was predictable. But it was honest and open. When people are smiling to mask their real interests it doesn't really help you," he said. (Additional reporting by Matt Robinson in Tbilisi; editing by Jon Hemming)

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