What is the difference between using Terminal Services Client and Remote Desktop in Windows to access another server?
"Terminal Services Client" is the old name/version of the program called Remote Desktop Connection in more recent versions of Windows.
Every Windows version beginning with Windows XP includes an installed Remote Desktop Connection (RDC) ("Terminal Services") client (mstsc.exe) whose version is determined by that of the operating system or last applied Windows Service Pack.
The Terminal Services server is supported as an official feature on Windows NT 4.0 Terminal Server Edition, Windows 2000 Server, all editions of Windows XP except Windows XP Home Edition, Windows Server 2003, Windows Home Server, on Windows Fundamentals for Legacy PCs, in Windows Vista Ultimate, Enterprise and Business editions, Windows Server 2008 and Windows Server 2008 R2 and on Windows 7 Professional and above.
The RDC 7.0 for example contains these improvements and new features:
- Multiple monitor support
- Web Single Sign-On (SSO) and Web forms-based authentication
- Access to personal virtual desktops by using RD Connection Broker
- Access to virtual desktop pools by using RD Connection Broker
- Status & disconnect system tray icon
- RD Gateway-based device redirection enforcement
- RD Gateway system and logon messages
- RD Gateway background authorization & authentication
- RD Gateway idle & session time-outs
- NAP remediation with RD Gateway
- Windows Media Player redirection
- Bidirectional audio
- Enhanced video playback
No differences from client side, it's even the different names of one product (I assume you not using TS client from early 200x). It allow to access both Terminal Services (on Windows Server family) and Remote Desktop Connection (Windows XP/Vista/W7)