Is there any way to hide taskbar labels and not combine taskbar buttons?

It's something of a hack job, but this might do the trick! It requires doing a bit of registry editing to shrink the taskbar icons so that the labels auto-hide.

Set taskbar options

1. Open Taskbar properties.
2. In the "Taskbar Appearance" group, change the "Taskbar Button" option to "Never combine" if you don't want stacking.

Adding/editing the registry entry

1. Press Windows Key + R to open the run command
2. Type int "regedit" (without quotes) and press OK
3. Go to HKEY_CURRENT_USER -> Control Panel -> Desktop -> WindowMetrics
4. Find the "MinWidth" entry.
5. If it is not there, right click on an empty space and select New -> String Value. Name this entry "MinWidth" (without quotes)
6. Double click on the MinWidth entry. If you want just the buttons to show, set this to 38 if you're using small buttons, 52 if you're using large buttons (or 54 in Windows 8). Otherwise any value above 38 will work. If you set a value below 38, the buttons will behave oddly. If your main display (on Windows 8.1 or later) has non-standard DPI (e.g. it's Retina or 4k display) you might need to multiply these numbers by the factor of 1.25, 1.5 or 2. Otherwise buttons will shrink / animate incorrectly.
7. Log off and log back in or restart to see the changes.

Deleting/Uninstalling

1. Press Windows Key + R to open the run command
2. Type int "regedit" (without quotes) and press OK
3. Go to HKEY_CURRENT_USER -> Control Panel -> Desktop -> WindowMetrics
4. Find the "MinWidth" entry and delete it.
5. Log off and log back in or restart to see the changes.

This is an old question, but the answer of Kyle B. didn't work well for me (glitches and flickering), so I'd like to propose an alternative solution.

1. Configure your taskbar for Always combine, hide labels.
2. Download 7+ Taskbar Tweaker.
3. Select the Don't combine grouped buttons option.

The end result:

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As a bonus, 7+ Taskbar Tweaker also allows you to configure this option per application: