Certificates from Let’s Encrypt will be trusted by most of the software (such as web browsers) likely to access your NAS without you needing to specifically tell the browser to trust that certificate. This is because the browser vendors trust Let’s Encrypt (actually, IdenTrust, who in turn trust Let’s Encrypt) to issue trustworthy certificates.
This is mostly a benefit if using many different devices to access the NAS, or if you need several other people (family, visitors?) to access the NAS. For a single person or a small number of people it may be perfectly convenient to explicitly tell the web browser that you trust your existing self-signed certificate, or even to just ignore any warnings and proceed.
The technology (e.g. encryption strength) used to access the NAS will be identical regardless of whether a Let’s Encrypt certificate or the self-signed certificate are used. However, there can be a risk that someone with access to your network (including a WiFi network if you use one) would be able to pretend to be the NAS. This will make them the “Man in the middle” able to spy on everything you do with the NAS, and change what you see if they wish. The warnings that enable you to spot such an imposter would be the exact same type of warnings you’ve described now, so having the Let’s Encrypt certificate (which would mean no warnings normally) might ensure you’re alerted to such an attack. On the other hand you may judge that your network is secure or that this is an unlikely state of affairs.
If you sometimes access the NAS from outside your home, the risk of someone impersonating it is obviously higher, bad guys are unlikely to stake out your house, but they might plant a device at a coffee shop or other popular browsing location.
Be aware that if a Let’s Encrypt certificate does sound like something you want, Let’s Encrypt can only issue you a certificate for a public Internet DNS name you really control, this is the sort of reason they’re considered trustworthy in the first place. If your NAS has such a name, even if it’s not normally accessible from the Internet, you can get a certificate. But if you don’t own such a name, you won’t be able to get any certificates from Let’s Encrypt.