Limitations of Tor
Tails makes it clear that you are using Tor and probably Tails
Everything you do on the Internet from Tails goes through the Tor network.
Tor and Tails don't protect you by making you look like any random Internet user, but by making all Tor and Tails users look the same. It becomes impossible to know who is who among them.
Your Internet service provider (ISP) and local network can see that you connect to the Tor network. They still cannot know what sites you visit. To hide that you connect to Tor, you can use a Tor bridge.
The sites that you visit can know that you are using Tor, because the list of exit nodes of the Tor network is public.
Parental controls, Internet service providers, and countries with heavy censorship can identify and block connections to the Tor network that don't use Tor bridges.
Many websites ask you to solve a CAPTCHA or block access from the Tor network.
Exit nodes can intercept traffic to the destination server
Tor hides your location from destination servers, but it does not encrypt all your communication. The last relay of a Tor circuit, called the exit node, establishes the actual connection to the destination server. This last step can be unencrypted, for example, if you connect to a website using HTTP instead of HTTPS.
The exit node can:
Observe your traffic. That is why Tor Browser and Tails include tools to encrypt the connection between the exit node and the destination server, whenever possible.
Pretend to be the destination server, a technique known as machine-in-the-middle attack (MitM). That is why you should pay even more attention to the security warnings in Tor Browser. If you get such a warning, use the New Identity feature of Tor Browser to change exit node.
To learn more about what information is available to someone observing the different parts of a Tor circuit, see the interactive graphics at Tor FAQ: Can exit nodes eavesdrop on communications?.
Tor exit nodes have been used in the past to collect sensitive information from unencrypted connections. Malicious exit nodes are regularly identified and removed from the Tor network. For an example, see Ars Technica: Security expert used Tor to collect government e-mail passwords.
Adversaries watching both ends of a Tor circuit could identify users
An adversary, who could control the 3 relays in a circuit, could deanonymize Tor users.
Tor chooses 3 relays that belong to 3 different network operators for each circuit.
Tails takes extra measures to use different circuits for different applications.
The Tor network has more than 6 000 relays. Organizations running Tor relays include universities like the MIT, activist groups like Riseup, nonprofits like Derechos Digitales, Internet hosting companies like Private Internet Access, and so on. The huge diversity of people and organizations running Tor relays makes it more secure and more sustainable.
A powerful adversary, who could analyze the timing and shape of the traffic entering and exiting the Tor network, might be able to deanonymize Tor users. These attacks are called end-to-end correlation attacks, because the attacker has to observe both ends of a Tor circuit at the same time.
No anonymity network used for rapid connections, like browsing the web or instant messaging, can protect 100% from end-to-end correlation attacks. In this case, VPNs (Virtual Private Networks) are less secure than Tor, because they do not use 3 independent relays.
End-to-end correlation attacks have been studied in research papers, but we don't know of any actual use to deanonymize Tor users. For an example, see Murdoch and Zieliński: Sampled Traffic Analysis by Internet-Exchange-Level Adversaries.