I agree with Andrew. I would just add that the first phrase is much more common than the second. The first phrase can refer to your house, apartment,, etc. The second phrase sounds more philosophical somehow.

Hi Pelin, The first phrase uses "live" as an infinitive, the "to" plus the base form of the verb. So the "place" is where the action of living happens. The second phrase uses the verb "live" as a noun by adding -ing to it to create "living." When -ing is added to a verb to create a noun we call it a gerund. In the second example, "living" is the object of the preposition to, and objects are always nouns. So, then, it is a place for the activity of living to exist. "Living" is understood to be a thing, not an action. I hope this helps. Happy English learning!

Infinitives excel at stating facts free of adornment. "A place (in which) to live" is a simple pure statement of the purpose of the place. Present participles create images. "Living" gives us an image of the ongoing action of a life. "A place for living" describes the activity for which that the place is intended.