What is the biggest difference between Dynatrace and Splunk?

Dynatrace and Splunk are two different solutions that provide insight from different perspectives. Dynatrace is installed on servers/applications to provide APM and Splunk is more a log and data analytics solution that processes logs (traps/syslog/Windows Events/Firewall Logs/etc.).

Dynatrace provides application performance management (APM), artificial intelligence for operations (AIOps), cloud infrastructure monitoring, and digital experience management (DEM), with products for the information technology departments and digital business owners of medium and large businesses. The company's services include performance management software for programs running on-premises and in the cloud. This software manages the availability and performance of software applications and the impact on user experience in the form of deep transaction tracing, synthetic monitoring, real user monitoring, and network monitoring.

Splunk (the product) captures, indexes, and correlates real-time data in a searchable repository from which it can generate graphs, reports, alerts, dashboards, and visualizations.
Splunk makes machine data accessible across an organization by identifying data patterns, providing metrics, diagnosing problems, and providing intelligence for business operations. Splunk is a horizontal technology used for application management, security and compliance, as well as business and web analytics.

Ref - from experience, their own websites and other related sites.

The two things are entirely different.

Splunk is primarily a log collection, analysis, and visualization solution. It can collect metrics now as well. Its purpose is after the fact forensics (what happened) as a part of a problem resolution process. That problem can span the entire gamut from security, to infrastructure operations to application operations. The primary competitors to Splunk are Elastic and Sumologic.

Dynatrace is an Application Performance Management solution designed to automatically measure the performance of an application (or a micro-service), discover the topology and dependencies that the application relies upon, and determine if a problem is in the code or in the software and hardware infrastructure that supports the application. The key to Dynatrace's ability to do this is its real-time topology and dependency mapping engine called SmartScape. There is no similar capability in Splunk. The principal competitors to Dynatrace are AppDynamics, New Relic, and Instana.

Rather than viewing these things as competitors, many companies use them together. In fact, Dynatrace has integration with Splunk, and Splunk has a Splunk App for Dynatrace up in Splunkbase. The most common use case for using them together is that Dynatrace finds the problem, determines if it is the code or not and if not determines where in the software and hardware infrastructure the problem resides. Splunk is then used to drill down into the part of the identified infrastructure to determine the exact nature of the problem (for example a security breach).

As far as I know, Splunk is used as a syslog system only and DynaTrace is an application monitoring system. So you will use Splunk to collect events/logs from servers and devices and you will use DynaTrace to monitor applications and send alerts to end users, provide dashboards etc.

Dynatrace is for APM / ITOM solution. Dynatrace trace can inspect the .net, Java, PHP Program. Splunk is for Log management and SIEM which Dynatrace does not focus on.
They are two different function products.

We use Dynatrace for deep dive transaction tracking to find performance issues of applications. We use Splunk various log analysis tasks mainly in the area of security.

It really depends on the use case. Dynatrace can actually enrich the data collected by Splunk and Dynatrace has an out of the box connection to Splunk. If the goal is application performance and end-user analytics then Dynatrace is the better solution. If the goal is security and threat analytics then Splunk is the right solution.

Splunk and Dynatrace are two different solutions. Most organizations use both of them. Splunk can aggregate logs from Dynatrace. It also depends on what is the purpose of the usage. If you intend to measure end to end application performance and the application logs are instrumenting the metrics, then Splunk alone can do the job. It also allows you to correlate other events like firewall, network and other dependent applications/services.