"Ten budynek jest niższy niż tamta wieża."

Translation: This building is lower than that tower.

[AdamMlodoz]

When do you use niz and when do you use ode?

[Vengir]

„Do” has completely different meaning. Simply speaking it means "to/towards".

„Niż”, beside being also a meteorologic term, is used like English "than" in comparisons like the one above.

„Od/ode” is basically "from", but it is interchangeable with „niż” in these kinds of comparisons. The use of „od” or „ode” depends on the following word. Basically, if the resulting cluster would be too difficult with „od”, we use „ode” instead. It behaves exactly like „w/we” or „z/ze”.

EDIT: Clarification: After a quick query I only found two relevant uses of „ode”.
One is „ode mnie”. The other is in Lord's Prayer: „ode złego” (standard modern Polish would say „od złego” instead).

[immery]

One important thing is that :

od needs genitve
niż needs nominative

they are interchangeable in this kind of sentences and I think only it's mostly personal preference. I always use "od". I'm pretty sure most learners prefer niż :)

[Jack.Elliot]

This building is smaller than that tower

[Jellei]

"lower" is the main answer. Well, I'm not so sure that "smaller" is the same thing. After all, the White House is lower than the Tower of Pisa, but it's rather bigger taking into consideration all the surface...