Why logging to a file when you could just set the HITSIZE variable in your .bashrc ? (plus this will give you ctrl+r search which is a must)
So, basically, in combination with sp332's and raldi's points, the answer is that it is completely not the same.
## SANE HISTORY DEFAULTS ##
# Append to the history file, don't overwrite it
shopt -s histappend
# Save multi-line commands as one command
shopt -s cmdhist
# Record each line as it gets issued
PROMPT_COMMAND='history -a'
# Huge history. Doesn't appear to slow things down, so why not?
HISTSIZE=500000
HISTFILESIZE=100000
# Avoid duplicate entries
HISTCONTROL="erasedups:ignoreboth"
# Don't record some commands
export HISTIGNORE="&:[ ]*:exit:ls:bg:fg:history"
# Useful timestamp format
HISTTIMEFORMAT='%F %T '