"大きそう and 大きい見える both mean that something looks/appears big. What is the difference between them?"
For starters, 大きそう is grammatical, while 大きい見える isn't. :) The latter would have to be 大きく見える to work: since the adjective 大きい is immediately followed by a verb, the adjective ending in い must change to the adverb ending in く.
大きそう often doesn't really mean "it looks big" in the literal sense of "look" as something's visual appearance. If something is just plain old big, folks will generally just say 大きい. The -そう on an adjective stem without the final い comes across more as "it seems like ..." So 大きそう comes across more as "X seems big." This might imply a relative comparison (which is probably the connotation of the uses that Chocolate mentions).
(Note also that -そう after an adjective that includes the final い indicates reported speech -- "I've heard that XX is YY; apparently XX is YY". So 大きいそう would be "[someone told me] that XX is big.")
Whether to use -そう or -く見える also depends on the grammatical context: what comes next in the sentence? If someone is trying to say "X seems big, but it isn't really", you might run across something like the following:
Ōkisō ni omoeru kedo, jitsu wa sasai na koto da.
It might seem big, but actually it's just a minor thing.
Alternately, you could say:
Ōkiku mieru kedo, jitsu wa sasai na koto da.
It might look big, but actually it's just a minor thing.
Note also that 見える implies something visible. If you say おいしそう, it might be that something looks delicious, or sounds delicious, but if you say おいしく見える, you're saying that something specifically looks delicious.