In addition to the above, the Ticonderoga is a large vessel suited for open ocean, while the Burke is more of a littoral combatant.
The important feature of both vessels was AEGIS, which is a missile system capable of engaging over 50 targets at once. In emergencies, a single button press would fire a missile at any detected target not confirmed friendly using a volley of launches from the Vertical Launch System (VLS).
Note however that early Ticonderogas did not have VLS and instead had a single SM-2 mount, without this volley fire capability:
AEGIS was in response to the highly capable Soviet bombers coming out, with AS-4 and AS-16 air to surface missiles, which would overwhelm US fleets with huge volleys of these weapons. These missiles were not trivial to jam as the earlier Styx missiles used in Arab-Israeli conflicts and would have to be shot down using a long range missile more deadly than earlier SAMs such as Terrier.
AEGIS was designed to data link with the Hawkeye, Nimitz class, and F-14 Tomcat systems then coming out, to provide comprehensive air defense of the fleet against these new threats.
Intercepting a Soviet Tu-95 Bear:
The E-2 Hawkeyes would provide an Airborne Warning And Control System (AWACS) which would alert the rest of the fleet. Tomcats would scramble to intercept the bombers. AEGIS would down any missiles that got through. The Nimitz carriers would coordinate everything through their onboard Combat Information Center (CIC). These systems would all network together to defend the fleet. AEGIS has its own CIC as well, in case it needs it.
Fortunately, this integrated fleet defense network has never seen major use.
The Arleigh Burke was introduced after the Ticonderoga class. It was an upgraded version incorporating stealth features, such as a radar deflecting design and radar absorbment materials.
The Burke has a CIC. It is placed lower on the waterline than the Ticonderoga, and is thus more protected from an above-waterline missile hit.
The Alreigh Burke is being upgraded to carry SM-6 missiles, which Raytheon claims is capable of intercepting tactical balistic missiles such as the SCUD.
The Alreigh Burke and Ticonderoga carry the same weaponry. The Ticonderoga is larger and suited for open ocean warfare. The Burke is suited for littoral combat in shallow waters, such as the Persian Gulf or Taiwan Strait.
Note that both ships have Tomahawk, Harpoon, ASROC, and 5 inch guns to engange surface (or subsurface, for ASROC) targets. The ships also carry an antisubmarine warfare helicopter carrying torpedoes, and Phalanx Close In Weapon System (CIWS) 20mm cannon as a last line of defense.