Is “drive them out” or “cast out” the same as “utterly destroy them”?

Answer:

Let's use Deuteronomy 7:1-2,16-24 since it has all the terms:

"When the LORD your God brings you into the land where you are entering to possess it, and clears away many nations before you, the Hittites and the Girgashites and the Amorites and the Canaanites and the Perizzites and the Hivites and the Jebusites, seven nations greater and stronger than you, and when the LORD your God delivers them before you and you defeat them, then you shall utterly destroy them. You shall make no covenant with them and show no favor to them. ... You shall consume all the peoples whom the LORD your God will deliver to you; your eye shall not pity them, nor shall you serve their gods, for that would be a snare to you. If you should say in your heart, 'These nations are greater than I; how can I dispossess them?' you shall not be afraid of them; you shall well remember what the LORD your God did to Pharaoh and to all Egypt: the great trials which your eyes saw and the signs and the wonders and the mighty hand and the outstretched arm by which the LORD your God brought you out. So shall the LORD your God do to all the peoples of whom you are afraid. Moreover, the LORD your God will send the hornet against them, until those who are left and hide themselves from you perish. You shall not dread them, for the LORD your God is in your midst, a great and awesome God. The LORD your God will clear away these nations before you little by little; you will not be able to put an end to them quickly, for the wild beasts would grow too numerous for you. But the LORD your God will deliver them before you, and will throw them into great confusion until they are destroyed. He will deliver their kings into your hand so that you will make their name perish from under heaven; no man will be able to stand before you until you have destroyed them."

When Israel took over the land of Canaan, it conquered the major strongholds initially but then gradually took over the remainder of the land for many years. This was to allow the land to still be cultivated until the Israelite population reached a point that it could manage all the land.

As they pushed the boundaries of the occupied land God would help in displacing the current residents. This would continue until these people cease to be separate nations. What would happen is that those who tried to stay would be killed, but those who left would be absorbed into the surrounding nations. For example, we have learned that the Hittites were once a mighty empire that was centered in Turkey. It started collapsing about the time the Israelites pushed into the land of Canaan. It so thoroughly collapsed that the world lost track of the Hittites until recent discoveries unearthed the buried cities. Israel did not kill all the people of the Hittites, but they were driven out of Canaan and eventually were destroyed as a nation.

Therefore, the terms don't mean the same thing but the casting out of the nations did lead to their utter destruction.