Both ARINC 429 and 1553B are communication standards for wiring and protocols widely used in civil and miltary aircraft applications.
During the early development of a spec suitable for aircraft use the civil designers judged a multiplexing databus to be too complex (cost, certification, compatibility with wide range of aircraft etc) and abandoned the 1553 development in favour of a simpler standard ARINC-419 and later ARINC-429.
The most significant difference (apart from speed and number of nodes possible) is that 1553 standard uses time-division multiplexing. This basically means multiple nodes - say sensors in the aircraft- can send their data "at the same time" along the same wire. (up to 31 at data speed up to 1M) It uses a command-response protocol.
ARINC 429 requires separate wires for transmit and receive at nodes (up to 20 data speed 100k), which means it's not a bus in the conventional sense as it requires more than one cable...
hugel