Lets Encrypt is a non-profit organization sponsored by EFF, Linux Foundation, and other major donors. Removing profit as a motive means their focus is security, stability, and open source, not their bank account.
It is fairly straightforward to totally automate certificates and renewals such that you never have to worry about the dreaded "certificate expired" error. There are numerous open-source implementations of clients to fit many situations.
Only class 1 certificates are available (the kind others charge $10 or more for). Lets Encrypt does not do Extended Validation or EV certs, nor does it offer insurance or the other features traditional vendors offer. How much of a con this is depends very much on how much you value those services.
Symantec is full-service, and that comes at a cost - their offerings are usually more expensive than the identical offering from their competition. A big part of what you are paying for is the name.
The 1 year (renewable) free TLS certificate does not allow wildcard certificate and limits to 5 host on a single domain. If you exceed that limit, you'll need to generate multiple TLS certificates.
In an unethical move, Comodo has attempted to register trademarks for "Let's Encrypt", a competing SSL certificate provider, claiming their "90 day business model" was stolen, despite Let's Encrypt being a non-profit CA.
Source: https://letsencrypt.org/2016/06/23/defending-our-brand.html
Your certificates will not be trusted by browsers until you install your private CA key into them. This makes this option unsuitable for public websites, but perhaps reasonable for securing private servers.
The fatal flaw of a Certificate Authority is pretending that they can be trusted. Individually, they can, but as a whole there are always CAs that are controlled by state actors or subverted by bad guys, and by using them you are betting that any abuse is discovered and handled quickly by the community. Running your own CA is placing trust only in yourself - you are not relying on a third party to vouch for a given certificate. That can be invaluable, or worthless, depending on your need.
Running your own CA costs you no money, but will cost you a certain amount of time and effort; moreso if you do it right. A good admin who knows what they are doing can set up in an hour or so, but you will also have the ongoing task of adding your CA to the trust store for the browsers you use.
While running a private CA is not difficult, doing so requires a fairly deep understanding of SSL and how CAs work. If you are not 100% sure how SSL actually works - this might be a bad option, or an awesome way to learn.