There is no 'right' answer to this question, as it totally depends on the needs of your specific store.

Shopify is an excellent choice if you want to get set up quickly and prefer to not deal with hosting, PCI Compliance and developers. They have many beautiful pre-made templates, excellent support and a huge marketplace that will provide you with just about any feature and function you need.

The downside to Shopify is that it gets extremely expensive. That $29/month plan looks great until you realize that you have to add another $100/month in apps to get it to do what you want. And Shopify Plus? That's $2000/month, which is insanity. As with most SaaS platforms, you also have very little control over the checkout, even with the Plus plan. So you may not be able to customize things as much as you would like. You also have to use Stripe if you don't want to pay extra transaction fees. Shopify also doesn't do multi-language, multi-currency or wholesale well, so if one of those is important to your business you will need to look elsewhere.

WooCommerce is a great platform if you want to be able to completely customize every aspect of your store. It is also free, and integrates seamlessly with WordPress, which is a bonus.

The downsides are numerous. You need a developer, there is no way around it. If you don't have a good development partner to keep your store updated and running smoothly you'll be in big trouble. There is also very little support (it is free, what do you expect?). You will constantly be dealing with upgrades and conflicts with other apps. PCI Compliance and security are constant concerns, as WordPress is the most hacked platform on the planet. You will need an excellent hosting partner as well, as that is the foundation you are building on.

Bottom line is, if you want a completely custom cart, Woo is a better choice but requires a developer, hosting and PCI Compliance. If you want simple and easy without having to deal with developers and hosting, Shopify is a better choice, but it comes at a steeper price.

Also don't limit your choices to these two carts - there are numerous other ones that may be a better fit for your business including Pinnacle Cart, Americommerce and Magento. They all have their pros and cons as well. Just because Shopify is the most popular doesn't mean it is the best for your business. Pick a cart that handles 75% of what you need out of the box, not just now but 5 years down the road as well. Re-platforming is never fun, so you'll want a cart that grows with your business. Good luck!