Distinguishing house from home

If you want to refer to home as in a special place, domus is the word to use, though there are also figurative options, like focus, meaning hearth, or by metonymy, home. I believe similar terms have been mentioned in another answer.

If you want to talk about a house, the word casa was used to mean house in Late and Medieval Latin, while in Classical Latin it meant hut, cottage, or cabin. Another answer mentions aedes, which works in Classical Latin.

Mansio means a dwelling, according to Lewis and Short, when used with a genitive:

II. Transf. (post-Aug.), a place of abode, a dwelling, habitation. A. In gen.: pecorum mansio, Plin. 18, 23, 53, § 194: aestivae, hibernae, vernae, auctumnales, Pall. 1, 9, 5; 1, 12: mansionem apud eum faciemus, Vulg. Joann. 14, 23: multae mansiones, id. ib. 14, 2.

Mansio also is a night quarters or inn, or, in the context of a journey, a stopping place/station.

Habitatio can mean dwelling, but also the rent for a dwelling.

Keep in mind, with habitatio and mansio, they're derived from the verbs habitare and manere; they still both retain their meanings as action nouns.

A word for “house” is probably the easy part:

• aedes, -ium, f. literally means “rooms.” Only fits if your dwelling has more than one room, although, if it has only one, you could call it aedis.
• domicilium and habitatium are pretty generic terms

However, I am partial to tectum (roof) in the metonymical sense, which I think emphasises the function of a living place to shelter you from the elements, and no more.

For “home” as the place where you belong and feel safe, I would suggest lar. It literally refers to the lares domestici, the Roman household gods that belonged to the house and often had a small shrine near the hearth, but metonymically also stands for someone's home. Lar familiaris seems to have been used in the sense of “familiar home,” e.g. Sallust quotes Catilina holding a speech about social inequality (that might as well have been held by a Frenchman in 1789 or a Russian in 1917):

Etenim quis mortalium, cui virile ingenium est, tolerare potest […] illos binas aut amplius domos continuare, nobis larem familiarem nusquam ullum esse?
What mortal with a manly character can bear it […] when they are joining two or more houses together, while we have nowhere a familiar hearth?

So if you wanted to say (for example after moving to a city): “I've found a place to live, but I haven't established a home yet.” – you could say: Tecto subii, sed etiam sine lare familiari sum.