're not mutually exclusive: Each is suited to particular situations.
Both ActiveX controls and Java applets and Beans support the "download once run many times" architecture. In the case of Java applets, the Java virtual machine (JVM) interprets the bytecode and controls access to system resources. The JVM removes the applets from the details of the host OS, and thus applets can run on any platform that supports the JVM. Most do. ActiveX controls, by contrast, have to register themselves with the Windows registry and execute as native Windows applications.
The trade-off is portability versus native look and performance. Any version of Java code is cross-platform as long as the JVM supports that platform. But Java's performance is less than stellar, and by default the security subsystem of Java does not allow access to native features such as file I/O and devices. Thus, Java is the best bet for those shops that have to support a hodgepodge of operating systems and processors and don't mind having to write an application around the security features.
While Java is the jack-of-all-trades, ActiveX is the master of one: Win32. ActiveX controls function exactly like native Windows applications and thus have access to all native features such as print queues, file I/O, and even memory. For security, ActiveX depends on Microsoft's Authenticode system (see the sidebar "ActiveX Security"). The use of the native features of Windows means that ActiveX controls run at native speed, typically faster than their Java counterparts. Moreover, most popular client/server tools, such as Visual Basic, Delphi, and PowerBuilder, now support ActiveX control development using the native languages of the tools. Finally, many Windows tools and applications can make use of ActiveX controls, so they aren't confined to your browser.
Which is for you? Consider them both for their strengths and their weaknesses. It's okay to leverage ActiveX controls for homogenous Windows applications, while Java is a finer fit for cross-platform situations. It's better than force-fitting applications in the name of a standard, or worse, a religious crusade.