Time: finally

やっと

Means that something happened at last. More rarely means “barely”.

Example

やっと論ろん文ぶんを書き終おわった。 I finally finished writing my thesis.

ついに

Means “finally”, “at last”. Something happened after working to accomplish the goal.

Example

ついに 新あたらしい地下鉄ちかてつが出で来き上あがった。 At last, the new subway was completed.