I've found that different groups tend to use one or the other preferentially when referring to the penis, but that there is no consistent extra meaning. On the other hand, when used to refer to a person, dick has a derogatory meaning.
A few other slang terms for the penis (from Wikisaurus,
Wikisaurusenis - Wiktionary):
bait and tackle
boner (Refers to erection)
choad (penis with greater girth than length)
chubby (Refers to erection)
cock
Crotch Pickle
dick
ding-a-ling
ding-dong
dingy (childish)
dinkle
dong
dork
endowment
fascinum
flander doodle
fuck stick
fun gun
the fury
gherkin
hammer
hampton (Cockney rhyming slang, "W:Hampton Wick"="prick")
hose (fire, etc)
knob
love gun (popularised by the band Kiss)
love pump
john thomas (popularised by Lady Chatterley's Lover)
johnson
joystick
junk
lad (Ireland)
Light saber
love muscle ("Russell the")
main vein
manhood
meat puppet
member
Mr. Happy (As sung by 'Weird Al' Yankovic)
Mr. Jim and the Twins
Mr. Winky (Dr. Winky or just Winky in less formal settings)
one-eyed trouser snake
organ
package
pecker
peener
pee-pee (childish)
pee-wee (childish)
percy
peter
pickle
pink oboe
piny tenis "code" for tiny penis
pocket rocket
pole (pink, purple, sex, etc)
Pork Sword
prick
pud
purple-headed womb broom
purple-headed yogurt slinger
ram rod (Roger)
red headed stranger
Rumple Foreskin
sausage
shaft
shit pole
shlong
spirit stick
staff (chief of)
stiffy (Refers to erection)
skin flute
tallywhacker
thing
todger (Australia, UK also tadger)
tonker (From Terry Pratchett's "Lords and Ladies")
tool
trouser snake
tube steak
twig and berries
unit
wang
wee-wee (childish)
wiener (childish)
wife's best friend
willie
winkle
Woody (Refers to erection)
Stuart (Refers to ROD! stewart)