I think the biggest difference between the two is going to be batteries, but digital altimeters are going to come with a lot of fun features like tracking your elevation ascent and descent rates, and warning you when a potential storm is rolling in.
What a lot of alpinists carry with them for calculating altitude in the back country these days is a watch with a altimeter.
Barometric altimeters are more accurate for calculating exact elevation, but unless you're a cartographer then I don't think it's really necessity to know your elevation accurate to +/-1m. Lots of people just use their GPS device, what precision you get will depend on the quality of your device, how fine a reading it's capable of taking, and what angle the satellites are to the horizon, but the accuracy is typically between +/- 5m and 15m. If you're like me, and don't necessarily need to know an uber-accurate elevation, then you'll probably be happy just using the GPS in your smart phone. I suggest you get a backcountry navigation app that also tells you your altitude. I recommend you try out Backcountry Navigator before you spend big money on a new altimeter or GPS. I'm pretty satisfied with it, and it does a lot more than just tell you your elevation, it also shows you where in the world you are on a topographical map, maps your trail, and lets you set waypoints with pictures or selfies attached to them.