If you do the latter, you need not forgo the help a real estate professional. Two professional groups exist to help create this new fee-for-service marketplace. Here's a link to the section on one site explaining why consumers like you may like this unbundled service, money-saving alternative:
If you prefer, the two and a half minute video is worth viewing, too!
In fact, your opinion about your house probably doesn't matter.
You need the help of a Realtor, doing a CMA on both houses:
Have you been inside the neighbor's house?
There must be some differences.
The CMA will show Market Value of your house; if their house is prices much higher, it should help you sell yours.
That is a good thing.
What your neighbor listed their house for, could simply be what they "want" to sell it for, not what it "will" ultimately sell for. Unfortunately, there are many agents who let the owners dictate what the property lists for - and do them a large disservice at the end of the day with a lot of valuable time and energy wasted.
You'll need to be honest with yourself and look at your property objectively, from a buyers perspective. You'll also need to know the prices of what has recently closed, and what you'll be competing with when your property hits the market. What you believe to be a big selling point and an upgrade - may not be to someone else. Honestly and understandably, most people overvalue their property and end up saying various things like "well my home has XYZ and the house across the street doesnt, etc etc." Being completely objective is often hard to do, but extremely important. I would contact a licensed, full service real estate professional to help you through the process and run a free comparative market analysis for you.
I'll also add, there are various routes you can take these days to sell your home. You can A. list it yourself for sale by owner, B. use a flat-fee MLS service, and "add-on" additional services as you see fit for extra fees, or C. use a licensed "full-service" real estate professional to walk you through the process from A-Z, taking care of 100% of the marketing, contract, and negotiation - providing full support every step of the way.
You'll have to determine what fits your situation best. A and B will most likely require you to pay upfront, regardless of if your property sells. Choice C is full-service and might "cost" a little more, however you'll only pay if your property sells, and the commission can generally always be negotiated. Being a real estate broker, I am biased toward choice C - only because I have seen endless clients try the FSBO/flat-fee route, pay hundreds/thousands of dollars upfront only to have their home not sell, and end up calling me anyway.
Hope this helps!