As you can see below, they are essentially synonyms. However, intoxicate has a higher frequency than inebriate* and in the sense of "exhilarate" is in my experience much more common, so I would not say that they are gleich geläufig.
*You can check that here:
tr.v.in·e·bri·at·ed, in·e·bri·at·ing, in·e·bri·ates
1. To make drunk; intoxicate.
2. To exhilarate or stupefy.
often as adjective inebriated
Make (someone) drunk; intoxicate:‘I got mildly inebriated’
v.in·tox·i·cat·ed, in·tox·i·cat·ing, in·tox·i·cates
a. To impair the physical and mental faculties of (a person) by means of alcohol or a drug or other chemical substance:served strong cocktails that intoxicated all the guests.
b. To damage physiologically by means of a chemical substance; poison:birds that were intoxicated by pesticides.
2. To stimulate or excite:"a man whom life intoxicates, who has no need of wine"(Anaïs Nin).
To cause impairment, stimulation, or excitement by or as if by use of a chemical substance:"The notion of Holy War is showing that it has not yet lost all its power to intoxicate and to inflame"(Conor Cruise O'Brien).
1usually as adjective intoxicated [with object] (of alcoholic drink or a drug) cause (someone) to lose control of their faculties or behaviour:
‘he was charged with operating a vehicle while intoxicated’
1.1Excite or exhilarate (someone):
‘he became intoxicated with his own power’
PS It would be easier to address your questions if you provided some context.