Here are the definitions:
A Trojan horse, or Trojan, is malware that appears to perform a desirable function for the user prior to run or install but instead facilitates unauthorized access of the user's computer system. "It is a harmful piece of software that looks legitimate. Users are typically tricked into loading and executing it on their systems"
A rootkit is software that enables continued privileged access to a computer, while actively hiding its presence from administrators by subverting standard operating system functionality or other applications. Typically, a hacker installs a rootkit on a computer after first obtaining user-level access, either by exploiting a known vulnerability or cracking a password. Once a rootkit is installed, it allows an attacker to mask the active intrusion and to gain privileged access to a computer by circumventing normal authentication and authorization mechanisms. Although rootkits can serve a variety of ends, they have gained notoriety primarily as malware, appropriating computing resources or stealing passwords without the knowledge of administrators and users of affected systems. Rootkits can target firmware, a hypervisor, the kernel or, most commonly, user-mode applications.
Hopes this help! Take care!