In American English, a lot of words are spelt with a single consonant plus "-ed", rather than two consonants as you often find in British English. Why isn't "plan" spelt with a single consonant?
As chrylis observes, there is no distinction in AE/BE spelling of the inflections of words ending in a stressed syllable like plan. The duplication consistently indicates the pronunciation.
The ‘American’ practice (which is no longer exclusively US) of declining to double these final consonants applies only to final consonants of unstressed syllables. Travel is inflected as traveling and traveled rather than travelling and travelled, but refer is inflected with duplication, referred and referring.
To an unsophisticated American eye, a spelling with unexpected duplication like travelling suggests that its user pronounces the syllable -vel- with with a 'short e' [ɛ] and stressed (like the composer Ravel) rather than with a schwa [ə] and unstressed.
I suspect that to an unsophisticated British eye, a spelling with unexpected non-duplication like traveling suggests that its user pronounces the syllable with a 'long e' [i:] and stressed (like reveal).
When forming past tenses with "-ed" or progressives with "-ing", final consonants are generally doubled in cases where the obscure rules of English pronunciation would change the vowel sound without them. "Plan" has a short "a" sound, and using two "n"s in "planned" and "planning" keeps it short. Otherwise, the past tense would be "planed" with a long "a", which is the past tense of "plane".