After the bombing the FBI moved very quickly to utilise the public communications revolution that is Twitter and mobile phone cameras to identify likely bombers. Commercial and police? CCTV footage would have also helped.
Two days after the bombings the inability (or unwillingness?) of the two brothers to hide and their shootouts with police spoke louder than any confessions would have that they were the bombers.
Most likely the brothers felt guilty that they were unable to fight or practice jihad (roughly translated as holy war) against the Russians in the ongoing insurgency in their place of birth, the Northern Caucasus which includes Chechnya, and the Russian republic of Dagestan . Both republics are mainly Sunni Muslim. The Russians have and are cracking down on Chechen "terrorists" much more severely than the FBI could in America (where human rights are a high priority).
Tamerlan, during his 6 month (January-July 2012) return to the Northern Caucasus, may have been aware that Russian security were on to him or that he decided to return to the US because life was too hard and risky there compared to congenial Boston.