I would translate "trotz" as "despite" or "in spite of", while "trotzdem" is more like "nonetheless" or "nevertheless". Moreover, "trotz" is a genitive preposition that is used before nouns, so you need to make sure that you use the correct article and form. Conversely, "trotzdem" is used with a verb.
For example:
Ich gehe trotz des schlechten Wetters spazieren. (I'm going for a walk despite the bad weather.)
Das Wetter ist schlecht. Ich gehe trotzdem spazieren. (The weather is bad. I'm going for a walk nevertheless.)
Ich will nicht, aber ich mache es trotzdem. (I don't want to, but I'm doing it nonetheless.)
