"I'm thinking to buy new running shoes. What are the pros and cons of their running shoes? What brand should I get?
Thanks in advance."

You should get the brand that fits your feet best.

Janis is 125% correct.

That said, between those two brands, Nike is usually less expensive and more light weight, while Adidas usually last longer (due to the type of foam and rubber used) and have a bit more cushion

A lot of people find Nike shoes on the narrow side, but many models have available widths. I'm a competitive sprinter and so my legs take a pounding and I like a lot of Adidas shoes to absorb impact. That said, I'm still a big fan of Nike, I just tend to use them for lighter workouts.

Source: 2 years in running specialty, operator of a running shoe review blog, competitive runner, former coach specializing in geriatric runners.

Also, don't bring brand new shoes on a walking intensive vacation unless you've worn the exact same model before (and even then maybe not). You don't want to discover that they give you blisters or other pains.

Neither of those brands fit me well.

I wear Asics, Ryka, or Saucony.

But it also depends the purpose. Are you actually "running"? As in training for a marathon? Or you just want a good walking shoe? Dance fitness? Tennis or racket sports?

Different shoes have different features (even within the same brand) that make them better for certain uses.

Another vote for Saucony. They are the only brand that seem to fit my feet properly.

So the answer to the OP's question is "whichever is more comfortable FOR YOU".

I used to always wear Saucony . . . but either my feet changed or their last did. You really need to try different brands/styles to figure out which works best for you . . .

Funny janisj... and I USED to wear Nike. For years, loved them, could count on them. But they currently don't work for me. Sneaker brands and style do change.

I have even tried to reorder exactly the same sneaker (Zappos) but they came in a different color (same brand same style name) & were no where near as great as the first pair I had.

So for our OP, you need to get out and try on some shoes. Wear them around. I like to shop at DSW because you pull your own shoes and try them on without a clerk there.

And as others mentioned, if this is for travel you want to have them at least a few months in advance of the trip to wear them around.

I think it's simply a matter of fit, and also style. But most importantly fit. I personally prefer Nke as their running shoes have a much thinner sole and I prefer it that way. Adidas tend to have bulkier sole.
Apart from that, be aware thhat a proper sport store can measure ur foot and suggest the shoe most appropriate for u in order to avoid injuries

I also wear Asics.. It adapts very well to the foot and also absorb impacts better than other

Depends on your feet. For my I wear only nike. Adidas only for causual.

nice sharing!!

Over the years I’ve worn Adidas, Nike, New Balance, Asics, Brooks and Merrells. They have all fit fine and my feet are certainly not perfect. Go with what is most comfortable, best fits your price range and your preference for design/ color. All things considered one brand is no better than the other.

Of forget that I’ve worn Sacony too. A particular style may incur small changes each year which may or may not affect fit.

I have two Saucony under my desk at work now. In theory they are the same shoe, purchased a couple years apart. Love one pair, the second pair not so much. They are supposed to be the same shoe, but they are NOT. I've had this happen trying to reorder other brands as well.

I prefer Nike because of the fit and not really about the looks. I just bought a pair of Under Armour shoes last month I find it very comfortable also, therefore maybe I'll start buying more from them.

I love the Nike Flex. Last trip where we went military cargo I thought we were headed to the UK but a flight came up for warm sunny Rota Spain. I headed to the exchange and bought Nike flex. I never had one blister and I walked 13 miles one day. I have run the Disney Marathon and Nike has the higher arch that I need. I can wear converse also. I think my feet were molded by them. I started wearing them when I was 13 many decades ago. The toe room is perfect in them and love my black on black converse but the Nike flex are my favs now. They dry really quickly.