Both ProtonMail and Tutanota are privacy-centric and secure email service providers. That said, I shall focus on their differences here.
Anonymity
Both Tutanota and ProtonMail heavily value user privacy and anonymity.
Tutanota logs IP addresses, but they are stored encrypted in the user's account, and can only be decrypted with the user's password. Tutanota also allows anonymous sign-ups, but subject to anti-spam measures (which may result in your anonymous account remaining unusable unless you contact Tutanota with another less anonymous e-mail address). They also don't accept any cryptocurrency, which is a nightmare for those who want to remain totally anonymous (where using Tor browser is a fundamental activity).
ProtonMail logs IP addresses and temporarily to prevent abuse; they claim those IP addresses are anonymized. ProtonMail also requires verification (through another e-mail address or phone number) upon signing up, unless you pay for premium immediately. However, when you do want to upgrade to premium, ProtonMail accepts cryptocurrency and cash envelope mail, along with conventional payment methods such as PayPal and credit card.
There is no clear winner. Tutanota wins when you decide to continue using a free account, ProtonMail wins when you want to pay for premium.
Encryption
Tutanota encrypts the entire email (except the sender and recipient e-mail addresses, which makes sense), though it does not support PGP ("Pretty Good Privacy") encryption supported by nearly all email clients. If users want to send end-to-end encrypted messages to non-Tutanota email addresses, Tutanota creates a temporary email addresses to the recipient(s) which can be read only with a password.
ProtonMail encrypts only the content of the email, not the subject, when sending to ProtonMail addresses. However, it supports PGP encryption which is compatible with most other email providers.
The winner here is Tutanota. However, some users may find PGP really necessary and would still prefer ProtonMail.
Jurisdiction
Tutanota is based in Germany, a 14 Eyes country. Germany is an ally of the United States which is the worst in privacy in all "free and democratic" countries. The German government and corporations will provide people's data to the US government when requested (and sometimes the other way around) when the German courts order it.
ProtonMail is based in Switzerland, which is a neutral country and has one of the best privacy laws in the world.
The clear winner here is ProtonMail.
Pricing
Tutanota provides 1 GB of free storage. They allow only a few emails per hour (based on their anti-spam algorithm) for free accounts.
ProtonMail provides 500 MB of free storage. They allow 150 messages per day for free acccounts.
Tutanota is slightly more favorable if you don't want to pay. However, 500 MB is enough for those who want to completely replace their less private email providers, unless they also want to store files there. Most other features are just more or less the same.
Both providers, on their premium plans, provide you with the ability to use custom domains, add email aliases, and add custom filters. Each premium plan comes with add-ons (such as additional aliases and storage size) that can be purchased separately.
Tutanota's cheapest premium option costs at least 1 euro per month. It allows you to ask help from their support, which is not provided to free accounts. However, upgrading to premium does not increase your storage capacity. (Upgrading your storage further from 1 GB to 10 GB costs at least an additional 2 euros per month.)
ProtonMail's cheapest premium option costs at least 4 US dollars a month. However, it increases your storage capacity from 0.5 GB to 5 GB. (Upgrading your storage from 5 GB to 10 GB costs at least an additional 5 dollars per month.)
It should be taken note that when paying for ProtonMail premium, using either the USD, EUR, or CHF currency does not change the absolute amount. For example, the premium monthly subscription costs either 5 USD, 5 EUR, or 5 CHF, ignoring foreign exchange rates.
While ProtonMail premium is more expensive, it comes with even more features than Tutanota. Regardless, Tutanota wins in both free features and premium pricing.
Customer support
Tutanota does not offer support for users having free accounts. You can still access their knowledgebase if you want to solve the problems yourself, or you can try asking the community in third-party platforms for support.
ProtonMail offers "limited" support for users having free accounts.
ProtonMail wins.
Interface
ProtonMail has a cozy interface, which shows you upon logging in a big banner asking you to upgrade to premium (if you haven't done it yet). It also shows another banner asking you to visit their online shop. However, ProtonMail has a lot more features which are visible just by looking on the home screen.
Tutanota has a clean interface, and does not immediately ask you to upgrade to premium (unless you stumble upon some of their premium-only features, for which you would see a pop-up asking you to upgrade to premium).
There is no clear winner. Tutanota would be more appreciated by non-technical users, while ProtonMail would be favored more by experienced users.
Conclusion
Tutanota wins in features (both free and paid). ProtonMail wins in jurisdiction. Both are roughly equal in terms of anonymity, encryption, and interface.
It is worth noting that ProtonMail already existed some time before the Edward Snowden revelations, while Tutanota came after. Therefore, it can be assumed that ProtonMail has much more experience in the field of secure and private email than Tutanota.
Based on my observations above, I could say that ProtonMail is generally better than Tutanota. There are still reasons to prefer Tutanota than ProtonMail.