I love taking photos of my family and our adventures but there's something different about video. It captures the sound of your child's voice, their funny little mannerisms and movements, and all the other little things a photo just doesn't capture in the same way. I love seeing photos of my own childhood but they don't compere to the home movies I have. Even the low quality, shaky, on-the-shoulder camera videos my Dad made just mean a lot more to me. It's like a time machine in a way the photo album just isn't for me.
The whole photo/video thing is a tough balance. You want to be present, be living in the moment and truly enjoying your time together but you also don't want to forget these things when your children are older.
If you can - I highly suggest getting a dedicated photo/video camera. Not only is the quality going to be better than your phone but it forces you to be more present. We bought a GoPro camera (this one) and I've been trying to shoot video whenever I can so I'll have these types of memories as she, and I, grow older. There are endless camera options out there but for me the GoPro is the perfect one for us. It's tiny (fits in your pocket / bag), it's relatively inexpensive and takes great quality video. It also is meant for sports/action so it does a good job of capturing a non-blurry video of the little one running around. You shoot your video or take your photo and keep moving. With a phone you take your photo, then most likely spend some time staring at your phone filtering, editing, posting, liking, commenting, sharing, etc. When you just shoot and edit/share later (back at the hotel, back home, at night after the little one is asleep, etc) it helps you be more 'there' -- at least for me anyway.
ANYWAY - We recently went on some really fun trips and I made a few short videos of them. Hope you enjoy them! Feel free to share your own videos or just your thoughts on the topic in the comments below.