Flash is designed for storing large amounts of data that are written en-bloc and infrequently - in the context of 8051s, the key application is to store your code

The trouble with Flash for data storage is that you can't just write to it; you have to erase it first - and you have to erase a whole block at once, not just individual bytes.
"Flash" would be more correctly known as "Flash-EPROM"; ie, Flash-Eraseable, Programmable Read-Only Memory (the "Flash" denotes the erase mechanism that facilitiates rapid, in-circuit erasing - invented by Intel, I think?)

For data storage (such as your ethernet packets) you need something with direct read and write access to individual storage locations.

SRAM is the simplest.

DRAM is cheaper per bit, but has the overhead of requiring a refresh controller. For the huge amounts (Gigabytes these days!) of RAM in PCs and the like, this overhead becomes insignificant - but for the kind of RAM size appropriate to an 8051 it's just pointless!

Stick with SRAM for your data storage