Why have people much more different cells than rice plants, while they have roughly the same number of genes?
Actually, rice has about twice as many genes than humans (you can check yourself on bionumbers). But who cares? It is not because two species have the same number of genes that they would look alike. Human and rice have quite different genes. Numbers in themselves do not matter.

Consider the situation where Paul has twice the number of clothes in his wardrobe than Jack, it does not necessarily mean that Paul and Jack dress up differently. Conversely, when playing poker, Paul always has the same number of cards in his hands than Jack, yet their hands may well be very different.

Generally speaking 'higher plants' tend to have way more genes than 'higher animals'. There are several hypotheses for that. Plants are more often polyploid and therefore have more neofunctionalization. Also, plants don't move. It is not impossible that they would need more genes to cope with a more varying environmental conditions.