These terms have come about due to the new technology available in vaping devices which are much more powerful than they used to be. If you are new to vaping, you will need to know the difference so you can make an informed choice about which may be best for you.
With mouth to lung (MTL/M2L), you inhale the vapour and collect it in your mouth before breathing it fully into your lungs. The atomisers used for this provide a tight draw and are what new vapers should start with first to ween off smoking.
The process closely mimics the effects of smoking, and is what early adopters of vaping will be used to as the first e-cigarette kits only allowed for the MTL method. Additionally, MTL setups usually require fewer power settings (wattage) and can handle higher nicotine and PG levels.
Direct to lung (DL) vaping is much closer to breathing normally. The DL devices are newer, more advanced setups and their atomisers have a much looser draw that require the use of your lungs directly to inhale.
Far more vapour is produced (cloud chasing), there is more flavour and the hit can be more intense and satisfying. Because of the bigger hit, it is important to choose a lower nicotine content (usually no more than 6mg). Users will also often favour a higher VG liquid which is much smoother, and a vape kit with airflow control.