There are two sides to this, building and electrics.
In terms of building yes, mindstorms can be seen as an extension of Technic and most of the Elements in a mindstorms set will be "Technic" elements. The old RCX was orientated towards "old-style" Technic that used a mixture of pins and studs while the newer NXT and EV3 are orientated towards new-style Technic that mostly uses pins.
In terms of Electrics things are more of a mess.
LEGO has had various electric systems over the years, broadly speaking there are.
- The 4.5V/12V system, this is ancient and completely incompatible with Mindstorms. This used plugs with two pins, sometimes with a central bump.
- The "electric system 9V", inroduced in 1990 and used for motorising many Technic sets. Also used by the RCX. This used contacts built into the top of lego studs.
- The NXT/EV3 system this uses a RJ style connector but with the clip in a different position from normal.
- The "Power functions" system, introduced in 2007 and has now mostly replaced the "electric system 9V". This uses a custom connector with two lego studs and a custom contact block next to them that is roughly 1x2 but won't stack with lego bricks.
Lego sold an adapter cable to go from the NXT to "electric system 9V" part number 8528. Unfortunately it is no longer sold. They also sell two different lengths of extension/adapter cables which can be used either to extend power functions or to connect power functions to "electric system 9V" part numbers 8886 and 8871.
It seems that Lego never made an adapter to go directly from the NXT to power functions though you can connect the NXT to power functions motors by combining the two adapters above.
Also it seems that the EV3 is not really compatible with the Lego adapter cable, apparently sensors won't work at all and motors will only work with special software hacks. There is discussion of this and possible workarounds with custom cables at How to use Power Functions with Mindstorms EV3? and Use legacy RCX sensors with EV3