I know there might not be a conclusive evidence about which technique/form-transformation is stronger, but how do they compare over all? Are there any features of one which are known to be superior to the other (speed, endurance, stree that puts in the body, etc.?)
Well firstly, this question can't be answered with complete certainty as we haven't seen Vegeta utilize it to its full extent. However, based on episode 122 and 123, SSJB Goku was shown to be stronger than SSJB Vegeta, probably cause of the Multiple Zenkai Boosts throughout the tournament of power.
Hence SSJB+Kaioken*20 Goku is a lot stronger compared to the time he initially used it against Jiren.
- Kaioken obviously puts a very heavy strain on the body. So even if Vegeta's new transformation happens to put a greater strain than SSJB, the probability of it straining the body more than Kaioken*20 is very unlikely. Hence Vegeta has the upper hand here.
- During the course of transformation, The Grand Priest stated that "Vegeta also broke through his shell". The also used here indicates that the grand priest is comparing it to Goku attaining UI. Based on episode 123, it's nowhere close to the power of UI Goku, however, based on the grand priest's statement and the spectators referring it to Vegeta's limit breaking power, I think it's fair to assume that it is going to be stronger than SSJB + Kaioken*20 Goku when used to its full extent.
- Finally, The Kaioken times 20 is just a 20 times multiplier. So to increase the power of the form, Goku will more or less have to increase his strength while utilizing SSJB. SSJB is all about ki control and utilizing God Ki. A superior form of SSJB must definitely have better Ki control and a better utilization of God ki. So Vegeta might very likely be able to get a lot stronger in his new form compared to Goku getting stronger in SSJB making Vegeta relatively stronger.
- Finally,this isn't a definitive factual evidence, however, SSJB + Kaioken is a powerup given to Goku in the Universe 6 vs 7 arc which was ages ago. The Universe Survival saga has seen a lot of characters being given massive powerups making them a lot stronger. It wouldn't make sense to give Vegeta a new transformation which is weaker than something Goku potentially unlocked a long time ago. Also, the ending shows Vegeta in his new form next to UI Goku, so I would think Vegeta's new form should indeed be superior to a form Goku attained a long time ago
Additional Proof: Spoiler
The provisional title of episode 126 is "Surpass Even a God! Vegeta's life-risking blow!". This could perhaps be hinting that Vegeta's new transformation is also God tier just like UI Goku.
A comment from Megumi Ishitami, director of that Dragon Ball Super episode regarding this issue
Vegeta's evolved version of Blue is an anime original. Like you said, that form is the result of him mastering his strength in a different way than Goku. Strength-wise please think of it as about the same as Goku's Blue Kaioken.
To be honest, my opinion is that SSBE Vegeta is stronger than SSB Kaioken Goku because Vegeta in that form fought more than Goku in the SSB Kaioken form. And Vegeta freakin' unlocked it in the end of T.O.P. Also, Vegeta in the manga said that Goku can keep his Ultra Instinct technique, also giving a smile while saying it. Here's a pic:
I rest my case.