A LOT :)
Sevgi is more broad, you can have sevgi for your mother, siblings, animals etc. AŞK is only for the one you fell in love with. Of course you can also have SEVGİ for your partner, but aşk is stronger and more passionate
The same goes for the verbs sevmek and aşık olmak, you might say "seviyorum ama aşık değilim" - difficult to translate :)
aşık olmak >> sevmek >> hoşlanmak :)
another interesting note, sevgi is of Turkic origin and aşk is of Arabic origin. Sometimes loan words take a slightly different meaning; so I wonder if aşk is also a strong passionate love in Arabic or if it just means love :)
I'm thinking that "seviyorum ama aşık değilim" is the Turkish equivalent of "I love him, but I'm not IN love with him." =)
Persian uses the Arabic loanword "eshgh" (same root as aşk) for romantic love and another Arabic loanword, "mohabbat" for familial love, so aşk is probably romantic love in Arabic too. :)
"Ashq" is definitively stronger in Arabic than just "hubb", which would be the equivalent of sevgi. From "hubb" we get "habibi", darling, maybe equivalent to canım.
If aşk means passionate love and sevmek implies a love that's more familial, then why do people say 'Seni seviyorum' instead of using 'aşık olmak'?
"aşık olmak" is "to fall in love."
"sevmek" can be romantic or familial, just like "to love" in English.
I'd like to point out a different aspect bu cümlede. Sevgi nedir = what love is. I've been wondering how to say something like that. A friend wrote on FB, "Mouse. It's what's for dinner," and ever since I've been wondering how that would be in Tk. "Fare: nedir akşam yemeği için." ??