Fairly tense. There's still ongoing culture war between the segment of the population who identify more strongly as Chinese (post-1949 arrivals and their descendants, primarily Kuomintang voters) and the descendants pre-1949 inhabitants who identify more strongly as Taiwanese.
The differences are subtle, but the greater exposure to Japanese culture due to the 1895 and 1945 rule is a factor, as is of course the distinct sense of isolation from being the mainland, which like any other island allowed a distinct cultural identity to develop.
When the Chiang Kai-shek's Nationalist KMT retrenched on the island in 1945 after the Japanese surrender, they viewed the Taiwanese culture with a certain amount of disdain for its subservience to Japan (many, including some Indigenous Taiwanese had even served in the Imperial Japanese Army during World War II) and expressions of this distinct culture was suppressed in public life.
Taiwan, meanwhile, had been developed by the Japanese to a greater extent than mainland China had been and the standard of living had risen with it, so there was a sense of simmering resentment on both sides and a very strong sense of Taiwanese civic life, especially when the KMT began taking control of core industries and handing out monopolies to loyalists so tobacco, sugar, tea, paper etc could be cheaply transported back to the mainland's crippled war economy. Inflation kicked in too.
Government and law enforcement was increasingly riddled with grizzled army officers, many uniformed freebooters and fanatics, who'd earned their stripes from 1927 in the Civil War and then in the Sino-Japanese War, and weren't perhaps the most delicate hands when it came to maintaining order and goodwill. In the ranks of the Chinese garrisons policing the island, violence and petty crime was rife. Tension was brewing.
On 27 February 1947 bout of brutality toward a black market Taiwanese cigarette vendor from a Chinese official from the Alcohol and Tobacco Monopoly Bureau lit the touchpaper on a vicious crackdown now referred to as the 228 Incident when a crowd gathered and the Monopoly agent fired into them, killing a bystander. On 28 February, Taiwanese civilians seized official buildings - protesters outside the Governor-General's residence were fired upon, inflaming the already volatile situation. Factions formed and demands were issued, unrest spread to the mainland where Communists pitched in to pour oil on the fire.
On 8 March, a significant Nationalist army arrived on Taiwan from the mainland and the crackdown began in earnest, with political opposition to the KMT rounded up and executed with the death toll seen as somewhere between 10,000 and 30,000. This isn't a comparison I've seen anywhere, but the way it's viewed is reminiscent as Katyn in that it's seen (by the Taiwanese) as the decapitation of the Taiwanese elite, the real cream of intellectual and political life.
US diplomat George Kerr's eyewitness account of the event, Formosa Betrayed, is available in full here
By the time the Nationalists had been forced from mainland China and onto Taiwan, the honeymoon period was well and truly over and the KMT were in a state of high anxiety.
A state of martial law was declared, ostensibly to protect Taiwan from the Communist forces across the water, but this provided a screen for combating any and all perceived challenges to Nationalist rule. Also known as the White Terror, which lasted from 19 May 1949 to 15 July 1987 and resulted in the imprisonment of around 140,000 Taiwanese people accused of opposing the KMT.
Sources: Taipei 228 Memorial Museum (I took lots of pictures of the displays, when I visited funding had just been withdrawn because KMT won the majority in the most recent election), Taiwan's 228 Incident and the Politics of Placing Blame by Craig A Smith and Formosa Betrayed by George Kerr. There's a woeful lack of books on the subject, frankly.
EDITED: For clarity and George Kerr.