What are the similarities between plant and animal cells, other than similar organelles or that they are both eukaryotic cells?

It's difficult to discuss similarities between the two cell types without mentioning their eukaryotic nature or their shared organelles, because these elements are so fundamental to their identity. However, we can generally talk about and compare their metabolisms, organization and lifestyles without turning it into a list of organelles.

Plants and animals both came from the same common ancestor, although we don't know exactly what that common ancestor was or when it lived. This is...

It's difficult to discuss similarities between the two cell types without mentioning their eukaryotic nature or their shared organelles, because these elements are so fundamental to their identity. However, we can generally talk about and compare their metabolisms, organization and lifestyles without turning it into a list of organelles.

Plants and animals both came from the same common ancestor, although we don't know exactly what that common ancestor was or when it lived. This is evidenced not only by our shared organelles, but by our similar metabolisms, DNA and appearance in the fossil record. We know that the fundamental structures of cells, such as phospholipid membranes and the structure and usage of DNA must have been a very ancient trait, because they're virtually identical in plants and animals, in addition to various reactions such as respiration. Basically, plants and animals were latecomers to the game of life, and took the foundations laid by billions of years of evolution and made themselves exponentially more complex, but the basic elements are clearly inherited from something much older.

Plants and animals are also basically the only things that can be confidently said to practice sexual reproduction, which is part of why they have the diversity and complexity that we have come to associate with them. Likewise, they are generally unable to alter their DNA during their lifetimes, as bacteria are. 

Finally, plants and animals tend to be highly organized, and in multicellular forms, they tend to have specialized sub-categories of cells that perform a particular function within the organism. Unlike bacteria or other single-celled organisms, the DNA contained in any given multicellular plant cell may not represent the entire genetic content expressed in that particular cell.