I see two things that may be causing a problem. But first, I want to suggest a few tips.
When you open a file, it is useful to use the /GET_LUN keyword because it allows IDL to find and use a logical unit number (LUN) that is available (e.g., in case you left LUN 3 open somewhere else). When you print formatted data, you should specify the total width and number of decimal places. It will make things easier because then you need not worry about changing spacings between numbers in a file.
So I would change your first set of code to the following (or some variant of the following):
OPENW,gunit,filename[0],/GET_LUN,ERROR=err
FOR j=0L, N_ELEMENTS(freq) - 1L DO BEGIN
PRINTF,gunit,freq[j],power[j],FORMAT='(2e20.12)'
ENDFOR
FREE_LUN,gunit ;; this is better than using the CLOSE routine
So the first potential issue I see is that if your variable power
was calculated using something like FFT.pro, then it will be a complex float or complex double, depending on the input and keywords used.
The second potential issue may be due to an incorrect format statement. You did not tell PRINTF
how many columns or rows to expect. It might not know how to handle the input properly, so it guesses and may result in those characters you show. Those characters may be spacing characters due to the vague format statement or the software you are using to look at the files (e.g., I would not recommend using Word to open text files, use a text editor).
Side Note: You can open and read the file you just wrote in a similar fashion to what I showed above, but changed to the following:
n = FILE_LINES(filename[0])
freq = DBLARR(n)
power = DBLARR(n)
OPENR,gunit,filename[0],/GET_LUN,ERROR=err
FOR j=0L, N_ELEMENTS(freq) - 1L DO BEGIN
READF,gunit,freq[j],power[j],FORMAT='(2e20.12)'
ENDFOR
FREE_LUN,gunit ;; this is better than using the CLOSE routine