近畿 is more specific. Being "near-capital" literally, it refers to an area including 大阪府, 京都府, 兵庫県, 奈良県, 滋賀県, 和歌山県, plus usually 三重県*1, seldom 福井県 (especially its southern part).
関西, on the other hand, has more variation in its meaning. In the narrowest sense, it is roughly the same area as 近畿. In its broadest sense, it can refer to the 'western' part of Japan including the 九州 area. Still, its centre is always considered to be 京阪神 area, and as such, it seems to be interchangeable to 近畿 most of the time (i.e. the broadest sense isn't very often seen). Dictionaries at my hand agrees that 関西, in recent usage, can refer to 京阪神 area, which is narrower than 近畿.
I also feel that 関西 has more connotation regarding culture, which may explain why the former is more frequently encountered. I don't think neither is more popular/common than the other.
Both are considered as 地方.
This Q&A from NDL can be an interesting reading.
*1: 三重 is tricky in this regard. It is also commonly included in 東海 area, and has possibly more connection to the 名古屋 city culture-wise.