Do they generally mean the same thing? Is one more formal than the other? Both 大陆, 中国 seem to have the same number of hits on Baidu.
When used with certain words, "中国" may be perceived more as a cultural/historical concept that has a vague boundary on what you want to refer to (the people, the land, the culture, etc).
Example: 周杰倫的中國風音樂／周杰伦的中国风音乐. Jay Chou's Chinese style music. (Jay is a popular artist from Taiwan.)
In this particular example, 中國 is similar to 中華 (The cultural concept). I think this is because 中国风 implies the usage of traditional style melodies, musical instruments, and lyrics, which is rooted in the cultural artifacts that have been existing for many centuries. The boundaries of "中国" in “中国风” feels vague, just like the bound of "China Proper" has kept changing over the many past centuries.
Another similar example: “中国话／中國話／中文”， the Chinese Language.
This may also happen in the phrase “中国历史”. The "中国" in this particular example would refer to the China encompassing both Mainland and Taiwan before Taiwan was ceded to Japan during the signing of《馬關條約》in year 1895, and/or the China during the Tang dynasty that did not include today's Tibet. The word "China" may not even just refer to a particular ethnic group, e.g. the Han ethnic group (that accounts for 95%+ of China's population as of today), and the Han has been practically ruled over several times during history. Examples are the fall of the Song dynasty in the 12th century and the fall of the Ming dynasty in the 17th century when the Manchus ruled. (However, the rulers began to use Chinese characters, which is an evidence that some say would justify that they were "assimilated into China", rather than "they annihilated China".) That is another example the word "China" may refer to a concept that has a vague ethnic boundary.
For "大陆", "中国大陆" and when "中国" is used independently today, there shall be an implied geographic bound in today's world, and the bound is usually much more clearly defined, although it may slightly differ on the person's political viewpoint. "中国大陆" and "大陆" are used (only) to refer to People's Republic of China founded in 1949.
A little digression below:
The word "中国" is said to have gained its new meaning --- a modern "country/sovereignty" in 1912 when Republic of China was established, as an abbreviation of 中華民國 (1). Prior to that, "中国" may more often mean "the central area/district/city" or the capital city. Example:
In short 大陆 = PRC, 中国大陆 = PRC, 中国 = at least PRC but probably includes "disputed territories" i.e. Taiwan. This is the accepted use of these three terms by the overwhelming majority of Chinese speakers.
By some Chinese, 中国 is used to refer to PRC + Taiwan + Tibet, territories that are historically Chinese. By referring to 中国, some speakers may be expressing their political view that China is not just the mainland but includes the aforementioned places. These people tend to be KMT (Kuomingtang) sympathizers. They will say 大陆 to differentiate PRC from 中国, which in their mind is a bigger country than the one run by CCP.
(Note: I'm mainly answering from personal experience here.) 大陆 is used to refer to "mainland China", which is effectively the PRC with the exclusion of areas such as Hong Kong, Macau, and Taiwan, which have (to varying extents) some political/legal distinction from what might commonly be considered China. 中国 has a more catch-all sense to it, though when used by Chinese speakers from areas not considered as part of 大陆, it can be effectively equivalent to 大陆.
It depends on where this native Chinese comes from.
Chinese from mainland China tend to use "中国" to refer to the China including mainland China, Hong Kong, Macau, and Taiwan. Chinese from Taiwan who claim to be only Taiwanese not Chinese, use "中国" to refer to China, a country that Taiwan does not belong to. Chinese from Taiwan who claim to be both Taiwanese and Chinese often use "中国" to refer to the China that the Republic of China used to reign, which is bigger than the current PRC.
"中国大陆" is mainland China for Taiwanese. Taiwanese will use "中国大陆" or "内地"（literally inner land） to describe where a Chinese from mainland comes from. They also use these terms in TV news or shows to describe the place where something happened. A Chinese from mainland will also use "中国大陆" or "大陆"（very rarely "内地" which is not proper） to describe where he/she comes from when he/she is talking to a Taiwanese. However, Chinese from mainland will not use "中国大陆" or "大陆" to describe the place where something happened when talking to other Chinese.
There are 4 parts: mainland (includes Tibet, Xinjiang and other provinces), Hong Kong, Macau and Taiwan.
In Taiwan, Hong Kong and Macau, 大陆 refers to mainland, which excludes Taiwan, Hong Kong and Macau.
People in mainland seldom use 大陆, they use 中国, which refers to mainland, Hong Kong, Macau and sometimes Taiwan.
People in Hong Kong, Macau, Taiwan, Tibet and Xinjiang think that they are NOT part of China.
People in mainland think that China includes Hong Kong, Macau, Taiwan, Tibet and Xinjiang.
What should you do?
In mainland, you should always use 中国, otherwise you will be pissed off.
In Hong Kong, Macau, Taiwan, Tibet and Xinjiang, you better use mainland when referring mainland.