It's a tricky one. Taiwan may be better for Westerners or people who are accommodated to ~Western~ societies, such that there is less "culture shock" at least initially. They use Traditional Chinese (can be a pro or con depending on how you look at it), and their accent is non-Standard. Taiwan people are greatly influenced by the Hokkien (闽南话; Minnan) language, which is different from Mandarin or Standard Chinese (普通话 ). Chinese teachers in Taiwan who have received professional training can probably speak Standard Chinese, but most locals speak in a non Standard way. Taiwanese also use a variety of romanisation systems plus Zhuyin / Mandarin phonetic symbols (注音符号), whereas China just uses Hanyu pinyin. 

For China it really depends where. First Tier cities are very modern and tidy. Actually, many Chinese who visit Taiwan are surprised how "backward" and "shabby" Taiwan is (buildings, roads, infrastructure, etc). The air in Taiwan is generally better than those in China though, so that might be something to consider. Inland China is generally less developed. But it really depends on what you are looking for, some people want to experience less developed societies. 

Places in Southern China don't use Standard accents. It's supposed to be areas in NE China (not Beijing or Tianjin) where the dialects are closest to Standard Chinese, but you may not be able to handle the Winter months, where it can get as cold as -20C. Learning Mandarin in China is more valuable to employers than learning Mandarin in Taiwan. So in terms of job prospects China is the better choice. You might also consider cities in East coast China like Qingdao (Shandong) and Shanghai but you have to be aware that the locals have their own dialects and accents there too (especially Shanghai where Wu (吴语) is the local language). That said, Shenzhen and Guangzhou are considered good cities to live in, so if speaking in a Standard way is not your priority then those cities are also places to consider. 

I read that there is also a recent surge in ~Hokkien~ and ~Native~ pride in Taiwan, so many people are using Hokkien rather than Mandarin to distance themselves from China. 

I was on about Harbin for -20C lol. Beijing and Tianjin don't use Standard Mandarin they have their own dialect (Beijinghua and Tianjinhua) which is not Standard Chinese, it is a heavy rhotic accent lol. I think Beijing can get cold but not as cold as Harbin. The average in winter is like -12C? Where I am now in UK, the average in winter is like 3C in winter, so it is comparably mild. 

But yeah in that case go for Taiwan. Taiwanese are said to be more friendly and warm to strangers and travellers. China in comparison is generally a bit more aloof and are less enthusiastic with people whom they don't know.

And yes the Chinese from China surprised about decaying apartments in Taiwan is true:



Taiwanese trying to be more "native":
en.wikipedia.org/wiki/Taiwanization

Yeah the /r/ sounds characteristic of Beijing and Tianjin area lol

And yes you could always study in Taiwan and take trips to Beijing or Shanghai or even remote parts of China for fun! Or even other parts of Asia like Singapore, Cambodia, Vietnam, Malaysia etc. 

If you speak Taiwan accent as a male, people in China think it's funny because it gives off an effeminate / flamboyant vibe. The analogy is the stereotypical gay accent in English. Taiwanese accent is close to the stereotypical gay accent in Chinese. On girls it's still acceptable but on men it's not desirable and people think it's funny. You'll be fine though because if you learn from an institution even in Taiwan, they should teach you the proper pronunciation, or at least close to it. 

Chinese find Taiwan "backwards" but it's mostly by those from the bigger and more developed cities. That's because major cities in China have very modern infrastructure and new buildings. It's like comparing the new city of Shenzhen (i.e. China's silicon valley) with Guangzhou (more traditional city).

Furthermore, I wouldn't worry about the cold if I were you. If you live in Ohio or Boston then it's not so bad and you can tolerate Beijing. I spent my formative years in Minnesota which definitely has harsher weather than all of Europe and most of China minus Russia/Siberia. If you go outside, away from heating haha

Even so, places like Harbin and Inner Mongolia are simply brutally cold when cold no matter what. One funny thing the climate in Northeast China is comparatively drier than Europe/North America hence Northeasterners fear a cold and wet climate, to the point of stereotyping South China as more unbearable than China's cold frontier.

I think they're being nice and pulling everyone's leg

You can also learn this in the West in better universities.
The teacher is usually an import from China or Taiwan.
This is a cheaper alternative but it works.
You can practice what you've studied in the nearby Chinatowns