Hey Grace, this is a great question - and one that I hear often.  Unfortunately, there's no clear-cut answer to this question, but I'll try to break it down into some digestable little nuggets to help you make sense of things.

When you're talking about DIY website builders (think GoDaddy, Weebly, Wix, etc), you'll be paying a recurring monthly fee, which will vary based on whatever bills and whistles you want/need.

Pro's

• Relatively easy to set up
• No web programming or design experience needed
• Short development time (often just a few hours of your time, if that)
• Depending on which service you go with, you'll have acessible tech support to help you with issues that may arise

Con's

• You don't really have a ton of customization over the end result, which means that your true brand personality might not shine through
• Other people's websites may look eerily similiar to yours
• You don't truly own the website, you're essentially just renting it.  If you stop paying the recurring monthly fee, your website disappears.
• Some companies have really lackluster tech support, and if something breaks, it may take a few days for someone to get to it

If you hire a developer or agency, you'll have an upfront, one-time fee, plus the costs of webhosting (which is relatively inexpensive).  Costs typically start around $500 for really basic websites, and go up from there.

Pro's

• You'll have a 100% unique website that allows you to express your true brand identity
• The end result, if you're working with a good company, will look exactly the way you want it to look
• The sky is the limit in terms of the number of bells and whistles that can be used
• No recurring monthly fees, you own your site outright

Con's

• Development time is longer
• If something breaks, you'll have to reach out for help, which typically won't be free
• You'll need to set up webhosting
• If you don't work with a good company, you'll find the process to be nightmarishly bad 

At the end of the day, it's all about preference, how quickly you need something up, and money.  If you can afford to pay for a website upfront, you'll end up saving a lot of money in the long run by working with an agency.  Plus, you'll own it outright.