Both ports are trunked.  All I did was enter "switchport mode trunk".  Here's the info you requested.

New Switch
VTP Version                     : running VTP2
Configuration Revision          : 1
Maximum VLANs supported locally : 255
Number of existing VLANs        : 5
VTP Operating Mode              : Client
VTP Domain Name                 : FHTM.com
VTP Pruning Mode                : Disabled
VTP V2 Mode                     : Enabled
VTP Traps Generation            : Disabled
MD5 digest                      : 0x23 0x7F 0x95 0x37 0xA6 0xA1 0xA3 0x0A

VLAN Name                             Status    Ports
---- -------------------------------- --------- -------------------------------
1    default                          active    Fa0/1, Fa0/2, Fa0/3, Fa0/4
                                                Fa0/5, Fa0/6, Fa0/7, Fa0/8
                                                Fa0/9, Fa0/10, Fa0/11, Fa0/12
                                                Fa0/13, Fa0/14, Fa0/15, Fa0/16
                                                Fa0/17, Fa0/18, Fa0/19, Fa0/20
                                                Fa0/21, Fa0/22, Fa0/23, Gi0/1
                                                Gi0/2
1002 fddi-default                     act/unsup
1003 trcrf-default                    act/unsup
1004 fddinet-default                  act/unsup
1005 trbrf-default                    act/unsup

Other Switch

VTP Version                     : running VTP2
Configuration Revision          : 5
Maximum VLANs supported locally : 255
Number of existing VLANs        : 10
VTP Operating Mode              : Client
VTP Domain Name                 : FHTM.com
VTP Pruning Mode                : Disabled
VTP V2 Mode                     : Enabled
VTP Traps Generation            : Disabled
MD5 digest                      : 0xE1 0x25 0x89 0xE4 0x8C 0xD8 0x3C 0xCA

VLAN Name                             Status    Ports
---- -------------------------------- --------- -------------------------------
1    default                          active    Fa0/1, Fa0/2, Fa0/3, Fa0/4
                                                Fa0/5, Fa0/6, Fa0/7, Fa0/8
                                                Fa0/9, Fa0/10, Fa0/11, Fa0/12
                                                Fa0/13, Fa0/14, Fa0/15, Fa0/16
                                                Fa0/17, Fa0/18, Fa0/19, Fa0/20
                                                Fa0/21, Fa0/22, Fa0/23, Gi0/1
111  VLAN0111                         active
222  VLAN0222                         active
333  VLAN0333                         active
444  VOICE                            active
555  GUEST                            active
1002 fddi-default                     act/unsup
1003 trcrf-default                    act/unsup
1004 fddinet-default                  act/unsup
1005 trbrf-default                    act/unsup

I actually forgot (about a year ago) when we setup the last switch we started using VTP passwords.  I think lack of sleep and memory was the issue on this one.  Thanks to everyone for all your help.  Much appreciated.

   Right it can't work the way it is because you have both switches setup as vtp clients.  You have to set the switch you want to create the vlans on as vtp mode server.  If the vtp domain names matches then  it should update the other  switch.  You can verify by looking at the vtp config revision number on both , they should match and all the vlans should match on both switches once it syncs.  That vtp config revision number will increment everytime you add or delete vlans on the vtp server, both server and client vtp config revision numbers should always match.

We currently have 6 switches on our network.  The VTP server is a 3560 and the rest are 2960s.  We did not setup a VTP password.  I am trying to trunk this new switch with another 2960.  The switch I am adding is our 7th on the network.

Just try add another VLAN to your 3560, and check if all other switches are getting updated. You can delete the vlan later. This addition and deletion usually triggers an update which causes your new switch to join the domain.

Here is the VTP Status from the 3560

VTP Version                     : running VTP2
Configuration Revision          : 5
Maximum VLANs supported locally : 1005
Number of existing VLANs        : 10
VTP Operating Mode              : Server
VTP Domain Name                 : FHTM.com
VTP Pruning Mode                : Disabled
VTP V2 Mode                     : Enabled
VTP Traps Generation            : Disabled
MD5 digest                      : 0xE1 0x25 0x89 0xE4 0x8C 0xD8 0x3C 0xCA

The VLANS

VLAN Name                             Status    Ports
---- -------------------------------- --------- -------------------------------
1    default                          active    Gi0/2
111  VLAN0111                         active    Fa0/1, Fa0/2, Fa0/3, Fa0/4
                                                Fa0/12
222  VLAN0222                         active    Fa0/5, Fa0/6, Fa0/7, Fa0/8
                                                Fa0/9, Fa0/10, Fa0/11
333  VLAN0333                         active    Fa0/16, Fa0/17, Fa0/18, Fa0/19
                                                Fa0/20, Fa0/21, Fa0/22, Gi0/1
444  VOICE                            active    Fa0/13, Fa0/14, Fa0/15, Fa0/16
                                                Fa0/17, Fa0/18, Fa0/19, Fa0/20
                                                Fa0/21, Fa0/22
555  GUEST                            active
1002 fddi-default                     act/unsup
1003 trcrf-default                    act/unsup
1004 fddinet-default                  act/unsup
1005 trbrf-default                    act/unsup

  Can you post the configs of the connecting  links for both switches and also a show int trunk .  Also reinput  the vtp domain name and make sure there are no spaces at the end of the vtp domain name when typed in .  It should work with what you are trying to do , the client should just pass the vtp advertisements thru that other switch.