It's no surprise that Princess Tutu has been called Utena-lite. Both series look like they're made for young girls and have, to some degree, fairly typical shoujo plots in the beginning. Soon, though, they become much more complex, twisting the definitions of friend and foe; of what it means to be yourself or to grow up. In addition, they're both heavily influenced by traditional fairy tales yet eventually change the norms of those tales to be something completely different.
Both Utena and Tutu contain a strong fairy tale motif and themes of accepting or defying one's fairy tale role. In a way, Utena feels like a more sophisticated/jaded exploration of the same themes touched upon in Tutu. They also both have a little bit of that surreal/reality-bending element, though Tutu never gets quite as surreal or symbolic as Utena. Basically, they're both really great series that you should watch, period.
Princess Tutu and Revolutionary Girl Utena have quite a bit in common. Both have the feel of a "modern fairy tale" and start off feeling like your typical, light-hearted shojo or magical girl series. However, both eventually obtain a darker atmosphere, become something new, and make themselves stand out in certain ways. Tutu is more family friendly and has more of a "magical girl" feel to it, while Utena is more intense and has more of a "girl warrior" feel to it. The heroines of both stories mature quite a bit throughout the story and learn concepts such as love, romance, friendship, self-sacrifice, and selflessness.
Both series have the feel of a modern fairytale. Despite being a more family friendly anime, Princess Tutu is not afraid to have stories as rich in symbolism as Utena.
The first recomender pretty much got it spot-on. Both are dark magical girl shows that, rather than focusing on the magical girl theme, focus more on the fairy-tale prince-princess theme. Utena is quite a bit more adult, however, and is a direct deconstruction of the Prince- Princess fairy tale genre, whereas Princess TuTu is just more of a darker 'Grimm' version of a fairy tale.
Utena is more psychological and explores more themes and has a more drastic use of metaphors.
Both series deal with the archetypes present in most fairy tales. While Princess Tutu explores the way the pure archetypes would interact in the real world, Utena looks at how these same archetypes would be either muddled or destroyed by the many shades of grey that exist around us, as compared to the black and white heroes and villains of a storybook.
Both of these series are shoujo based on fairy tales which invert the classic relationship between men and women in them. In each of these anime, the protagonist is a girl who takes on the role of the hero traditionally designated to the prince, a man. Both anime also ask if the heroic actions we see in fairy tales are really genuine and are really the right thing to do.
Two anime, heavily influenced by fairy tale stories, which turn into complex and twisted shows later on, revolving around similar themes, such as finding yourself or growing up.
Both of these shows are technically magical girl shows but they tackle serious and rather unexpected themes for the genre. Utena is more surrealistic and more difficult to unpack and often feels dark and heavy. It is very unique and an interesting experience nonetheless.