When comparing Jenkins vs TeamCity, the Slant community recommends TeamCity for most people. In the question“What are the best continuous integration tools?” TeamCity is ranked 1st while Jenkins is ranked 5th. The most important reason people chose TeamCity is:
TeamCity has different installation packages for different operating systems. All the user needs to do is download the correct one and run it.
Specs
Ranked in these QuestionsQuestion Ranking
Pros
Pro Highly customizable
Even though Jenkins is pretty functional and useful out of the box, there's a large plugin ecosystem from which the user can choose plugins to integrate into their Jenkins build. This is needed for when the user wants to extend any of the tool's features.
Pro Free and open source
Jenkins is a free and open source continuous integration tool, while its source code is hosted on GitHub.
Pro Safe to store key environment variables
Self-hosting provides a safe location to store key environment variables since it is the user who is in charge of the server and environment where Jenkins is hosted.
Pro A lot of resources and tutorials available
Jenkins has been in development since 2004 and is one of the most popular tools of its kind. This means that its technology is very mature and there is a lot of documentation and resources available for it.
Pro Multiple version control systems supported
Supports the most popular version control systems out of the box: SVN, Mercurial, and Git.
Pro Scalable
The distributed builds in Jenkins work effectively, thanks to the Master and Slave capabilities.
Pro Stable release line for users who want less changes
This is called the Jenkins Long-Term Support (LTS) version and helps to provide the most stable and assuring version of the Jenkins CI possible. Every 3 months, a version (which has been deemed the most reliable by the community) is chosen. After this, its branched, well-tested features are added (if they are missing), it is tested with the new features, bug fixes are then carried out if necessary, and from there it is released as the official Jenkins LTS version.
Pro User can source control their chain of automation
Starting with Jenkins 2.0, the pipeline capability, which has been available as a plugin before this version, has been built into Jenkins itself. This allows developers to describe their chain of automation in text form, which can be version controlled and put alongside the source tree.
Pro Quantity of available Plugins
For most operations we need not reinvent the wheel, there are plugins already existing.
Pro Cross-platform build support
Being a Java application it can be installed under any OS: Windows, Linux, and macOS. On the other hand, JNLP slaves also enriches the cross-platform build support for its agents.
Pro Easy to get up and running
A Jenkins install is very simple and the user can have the service up and running within minutes. To install Jenkins, the command java -jar jenkins.war
is all that is needed - nothing more.
Pro Supports most of the technological stacks for free by specific plugins
Including, Docker, Amazon EC2 and S3.
Pro Great community
Jenkins has a large and helpful community, which welcomes new users and provides a great number of tutorials.
Project Website, including links to Blog , Wiki, Docs. Community groups via Jenkins Users ML group.
Pro Self hosted
You stay in full control of your source code, build environment and deployment. No third party gets access to your source code or knows exactly how to build your software.
Pro Encryption of secrets
Thanks to JENKINS Credentials and Plugin.
Pro Multiple test environments for different runtime versions
They can be added easily under your Global Configuration.
Pro Awards and recognition
Including InfoWorld Bossie Award (Best of Open Source Software Award) in 2011, and Received Geek Choice Award in 2014.
Pro Easy installation
TeamCity has different installation packages for different operating systems. All the user needs to do is download the correct one and run it.
Pro Cross-platform build support
The fact that it is based on Java does not hinder TeamCity's ability to support different build environments. TeamCity in fact supports a large number of languages and tools for each of those languages (build runners and test frameworks).
Some of the languages/platforms that are supported include: Ruby, .NET, Java.
Pro Brilliant interface
The user interface of TeamCity is clear, well thought out and the dashboard is highly customizable.
Pro Supports build chains
The user can easily compose dependencies between builds by adding snapshot and artifact dependencies, all on the one screen. All output of upstream builds is available to downstream builds. Triggering sets off the entire build chain and supports re-running of the portions of the chain that failed.

Pro Well documented

Pro Extensible
TeamCity offers well defined APIs for extending, as well as a REST interface.
Pro Testing support
TeamCity supports both MSTest and NUnit (which is open source) to run tests.
Pro Best choice for .NET
Seems to be the best choice for .NET applications, but to be honest: if you stray from the default settings you will be in a lot of pain most of the time.
Cons

Con Poor quality plug-ins that are difficult to combine
There have been several complaints by users regarding the quality of the plug-ins found in Jenkins' official plugin repo. A lot of plugins found in the default plugin directory are no longer actively maintained and as a result, they may be incompatible with later versions of Jenkins or other plugins.
Con High overhead
Unlike some of the simple and hosted alternatives, users need to host and setup Jenkins by themselves. This results in both a high initial setup time, as well as time sunk into maintenance over a project's duration.
Con Unstable and lack of plugin integration QA process
Jenkins without plugins is almost useless. All plugins are treated equal and published almost right away.
Because there is no process for testing Jenkins' integration, the overall Jenkins experience is not that great. Furthermore, Jenkins' core and plugins are released on a regular basis, all requiring instant restarts, meaning that updates appear more than once a day!
Con Cannot get pipelines right
They are on the 5th attempt to make pipelines working. Still lot to be desired and clunky. Thoughtworks took CruiseControl and just rewrote it from scratch to make GoCD. Go.CD has pipeline support as first class citizens.
Con Restarting of pipeline steps only available in commercial version
Reliable pipelines with step restarts are only available in the enterprise version. Last time I talked to them in 2018, I was quoted $20k/year for that.
Con Vulnerable
Vulnerable to cross site and DOS attacks, read article Top 10 Java Vulnerabilities And How To Fix Them.
Con Limited pipeline size
Pipeline-as-code is limited to a JVM method size limit.
Con Expensive
TeamCity has a free tier which includes a maximum of 100 build configurations and up to 3 build agents. If you want to add 10 more configurations and 1 more agent, it will cost $299; unless you choose to buy an enterprise license which starts at $1999.
Con Poor quality plugins
At least some of them do not work, probably because they're not updated to more recent TeamCity versions.
Con Inter-branch merges trigger emails to unrelated committers
Whenever an inter-branch merge occurs, TeamCity pulls up the first parent of the merge commit and sends them an e-mail. However, this sort of information would be more useful to the merge author.
