I'm moving in two years to Yunnan. I'm learning Mandarin. But I've read that in Yunnan the official language is "Southwestern Mandarin".
What's the difference? Will they be able to understand me?
Southwestern Mandarin只在西南部分地区交流 当然只是语调的部分不同
The official language everywhere in Mainland China is just "Standardized Mandarin + other ethnic languages if applicable". So probably what they mean is that Southwestern Mandarin is the common language there. The difference is like British (RP) and American English to some extent. Since education is in Standard Mandarin, younger generations will understand you without problem. SW Mandarin is similar to Standard Mandarin, so generally older generations will understand you as well.
It is another issue whether you will understand them -- a perfectly fluent second-language American English speaker can often have difficulties understanding even American accents, and that will be similar to your situation. Younger people generally do speak SM, but if the other person happens to use SWM then I really do not know how that will sound to you. I personally have not had difficulty understanding SWM as a SM speaker, and navigated Chengdu and Chongqing without an issue (two major cities that nearly exclusively speak SWM), but that is in Sichuan instead of Yunnan, so I don't really know.
The places you go also matters. If you stay in the largest city, you will have considerably less problem than if you are going to live in a remote underdeveloped village.
Yunnanese Mandarin originated in the Ming Dynasty in the area called Nanzhili which is now today the provinces of Jiangsu and Anhui,were areas that spoke Classical Mandarin the language of the courts and officials from the mid 14th to 19th centuries and also the entire Yangtze and North China area, during the expedition south they brought this Language with them and the dialects spoken in Yunnan habe almost a Jiamghuai quality to them however 4 of the 5 tones of Classical Mandarin are preserved in Southwestern Mandarin, where as in the Central plains, Jianghuai and Northeastern dialects the vocabulary and tones have changed drastically, Southwestern Mandarin is actually the most representative in all Modern mandarin dialects of Classical Mandarin due to many of the tones and vocabulary being preserved which were lost in Northern China. If you listen to Peking opera you will notice they do not speak Modern Mandarin but classical Mandarin fron the Ming Dynasty based on the Honanese or Luoyang dialect of the time. As for Yunnanese there are aspects of it which match Jianghuai Mandarin due to most of the original settlers of Yunnan originating in Nanking.
Just use standard Mandarin, all of them would understand you without problem.
And there is no SWM in fact (as I live here for 20+ years). Yunnan is a mountainous province which is the perfect place for language diversity. The accent in different rural parts of Kunming (distance less than 40km) can lead to misunderstanding, not to mention the difference between accents of residents of Kunming and Dali.
But in general, all of those accent is similar to standard Mandarin. The only problem is elder residents would have problem pronouncing correctly in standard Mandarin, which might lead to a mixture of standard and localized pronunciation of Mandarin. But in general it is not a problem.