The majority of the differences are the architecture of the mcu's themselves. 8051 is the name of the architecture whereas PIC's use harvard architecture. Even so, when intel first produced the 8051 processor it was based around harvard architeture, the only significant difference that it should make at hobby level are those of the assembly commands. If you use a c compiler then even these dont make a difference.
Some of the 8051 type microcontrollers need you to add external eeproms etc to store the commands or program in wheres with PIC's everything is buit in such as logic, programmable eeprom, a small amount of working ram etc.. as well as built in analog converters and built in capture/compare ports.
You will probably find that PIC and AVR are easier to use but can seem more daunting at first because of additional functionalities.
ATMEL who also make microcontrollers and probably more commonly used on this forum also have some models of mcu based on the 8051 processor.