The most basic difference is they are two different size classes. Going by the EPA, the Camry was compact up until 1992, and midsized from then until now. The Corolla was a subcompact until 1993, and from then is considered a compact. Camry was always the next price point up from the Corolla. A Camry competed with cars like the Accord and Taurus while the Corolla did battle with the Civic, Cavalier, and contemporaries. The Camry is thus longer, wider, and offers more interior and trunk volume, requiring bigger 4 cylinders and offering optional V6s. This typically makes it a better cruiser and better for families at the cost of gas mileage, being harder to park, is more expensive. A used Camry typically has more options on it than a Corolla and also features a little higher quality materials and better NVH tuning. On the plus side a Corolla will be a little cheaper to operate, use less gas, and probably a little more nimble. Insurance may also be less.
Both are built to the same standards and aside from minor year to year differences you can expect similar reliability. For similar cars at the same price (in terms of year/mileage/features) I would go with a Camry. The Corolla has a bit of a premium on it in due to it's reputation as a reliable gas saver that doesn't seem to be abating despite gas going way down that tends to not always make them the best value as a used car. The Camry has a great reputation too but they seem to offer more for the money used. That said drive both and see what you like, but if you compare say a 2010 Corolla to a 2010 Camry, the Camry will probably be a little nicer car. The only ones I know to avoid are a run of 5th and maybe 6th gen Camry that had issues with the 2.4L, and from 98 to around 04 the 1.8L in the Corolla also had oil burning issues.