The phrase "in front of" implies that something is directly in front of you. "Across from" implies that there is a barrier in between you and someone or something else.
In your example, the park is across the street from the clinic. The street being the barrier between the park and the clinic.
"In front" would imply that it's directly in front of it ... nothing between it.
"Across from" usually means there is a road in between them.