Lax or lenient?

The words definitely overlap in their meaning and use, but looking at the definitions, lenient stays close to its primary application to mild punishment or discipline:

adjective

1.0 (Of a punishment or person in authority) more merciful or tolerant than expected:
in the view of the Court the sentence was too lenient
lenient magistrates

Origin

Mid 17th century (in sense 2): from Latin lenient- 'soothing', from the verb lenire, from lenis 'mild, gentle'.
_ODO

On the other hand, lax denotes a more general description of looseness applied to a wider variety of conditions and activities--muscles, speech, attention, expectations, discipline, morality, etc.:

adjective

1.0 Not sufficiently strict, severe, or careful:
lax security arrangements at the airport
he’d been a bit lax about discipline in school lately

2.0 (Of the limbs or muscles) relaxed:
muscles have more potential energy when they are stretched than when they are lax

2.1 (Of the bowels) loose.

2.2 Phonetics (Of a speech sound, especially a vowel) pronounced with the vocal muscles relaxed. The opposite of tense.

Origin

Late Middle English (in the sense 'loose', said of the bowels): from Latin laxus.
_ODO

Specifically, lenient usually connotes the soft treatment institutions, groups or individuals extend toward others, while lax is just as likely to refer to their relaxed general behavior, as in lax habits, operation and maintenance, which are rarely, if ever, used with lenient.

Even where lenient and lax overlap in usage, lax tends to have a broader application. In the corpus, the expression lax procedures is about seven times more common than lenient procedures:

More importantly, exploring the underlying data shows that lenient procedures almost always refers to procedural interaction with an outside entity, as exemplified in "lenient procedures of the German denazification" described in Nazi Crimes against Jews and German Post-War Justice, by Edith Raim:

The results of this lackadaisical approach soon became obvious. “It was pointed out that through the lenient procedures of the German denazification authorities and a consequent liberal employment policy by the Minister of Justice, a large part of formerly removed Nazis had been reinstated within the Württemberg-Baden justice system..."

In contrast, lax procedures are just as likely to refer to internal dynamics as external interactions, as seen in The Moral Responsibilities of Companies, by Chris Chapple :

The fine was, at least in part, intended to deter other banks from similarly lax procedures.

Conclusion

Although lax and lenient are synonyms that overlap in meaning and usage, the connotations of lenient focus more on mild punishment, discipline, and restrictions placed on outside parties. Lax generally describes a broader range of loose conditions and activities, and is not routinely limited to external interactions.