"Moved to the west" and "moved west" are indeed different. "Moved west", just means they moved westward to a non-specific location. It was just west of where they were before. The pioneers moved from the east coast and expanded out westward, in something called the "westward expansion".They went all places west of where they were - not just "the west".. An indefinite number of people moved to indefinite locations west of where they started. "Moved to the west", means it was a specific location. "the" is the key. For example. if someone in Asia moved to america they could say "She left Korea and moved to the west". "The west" being an accepted use of a term to define America/Europe as western countries. Likewise, I could say "I left America and moved to the far-east", this is an accepted term for regions Korea/Japan/Taiwan/China etc. If I am in America, and someone says "They moved west" I have no idea where they went other than it's just west of where they were. If someone says "The moved to the west" I would assume they meant California. But we would most likely say "They moved to the west coast", here I am left with no doubt they mean California. We also frequently say "she moved out west", this means likely anywhere in the "Pacific Timezone". These are just colloquial terms that americans have for places. The west Coast = california Pacific Northwest = Oregon / Washington

我的理解，既然是pioneers，他们并不知道自己要去哪里，并不知道要去某个确定的地方，只能有一个大体的方向，就是向西。所以，按照中国人的话来讲，第二个句子更有意境。Leaving their old lives behind, the pioneers moved west.(离开故土，一路西行)。

I think the previous answers are confusing West and west. When we talk about moving west or to the west, or out west, we are simply talking about moving to a place which is located in a more westerly direction - alternatively we might move north, east or northwest: it is easy geometry on the map. The West (capital letters) is a special place, but it means different things in different cultures. The West to most Americans is a place created by Hollywood in the 1940s and 1950s, a place of cowboys and Indians, rough justice, dramatic landscapes and battles between good and evil. When you read a history book about settlers in California, you should be careful not to confuse this with their direction of travel.

@Travis B, Thank you! I got it

assuming this takes place in America, The West is a place or region, like Texas or California.