Yes they are the same. Because of Chinese Character we(Korean) use, we often have two or more names. 

도쿄 is more like Tokyo's phonetic alphabet but actually, Korean says 도꾜 cuz 도꾜 is more close to Japanese

동경 is 東京's phonetic alphabet in Korean

도쿄 and 동경 has same meaning

but in official document, we write 도쿄

becuz that's the Korean grammer rule

I think this example is more easy to understand

cafe and café has same meaning right?

cafe is English word and café is French word

so 동경 is more like care and 도쿄 is more like café

I hope my explanation could help

동경 comes from Chinese word 东京

도규 comes from English word Tokyo

But Japanese and Chinese word for Tokyo are the same, just pronounced differently. Korean word 동경 uses hangeul to pronounce same word.

Another good example is France, it can be :

프랑스 comes from original pronounciation

법국 comes from chinese word 法国