I did not find any specific pros and cons between tab group and layout group, other than that Layout Group is better UI for mobile applications also. I hope this helps you:
The layout group allows you to specify the type of navigation a user will use to navigate sets of information. For example, this could be a tab for each account, or by simply changing the type, the account list can be rendered as either a drop-down menu, accordion, tabbed, or list. Additionally, a layout group is a way to present information responsively that is normally shown within one control-type.
A layout group can be rendered as a tab control, an accordion, a stacked format, or a drop-down menu according to screen size.
A layout group can contain any number of dynamic layouts, column layouts, repeating dynamic layouts, or other layout groups.
With a layout group, what would usually appear as a tab control on a regular desktop or laptop converts to a different navigation mechanism on a touch device, negating the need for horizontal scrolling. (for Mobile Applications)
Not necessary that we need to have any active tab. User can go ahead and collapse all the tabs by collapsing the layout group.
Basic difference is that Layout Group is responsive and device friendly.
Layout group comes with the responsive breakpoint settings(available in skin) which can be used to display the layout group in different formats based on the device width. This is something which is not available using the old Tab Group layouts.
Like Sathish said, Layout group is responsive and Tab is not. It's a good UI practice to design your screens so they render nicely on different form factors. Layout group used as Tab would allow you to do that. If you use 'Tabbed layout', the screens may be desktop friendly but on phones the screen will have horizontal scrollbars which results in bad user experience.
However, there are scenarios where Tabbed layout is superior and if you only cater to desktop applications there are certain advantages of Tabbed layout over layout group -
'Refresh when active' - You may set any tab to refresh when it becomes active only in Tabbed layout
Default active tab - You may set any tab (not just first) as default active tab
Tab position - Tab position may be changed from top to bottom, left or right
Cons of tabbed layout -
Styling is a painful process
Not mobile/touch friendly
No enhancement planned for future
A UI engineer like me would suggest using (and switching to) layout groups because that is where enhancements will happen. Tabbed layout should be considered only when you're designing applications for desktop.