Differences in MATLAB's versus Julia's beta(x, y) for large x or y?

Weird. From my side

julia> using SpecialFunctions

julia> gamma(big(3.6e13))
7.882861104028365451673058070377064242970022825665897715265241972922672893761775e+472392288679098

julia> @which gamma(big(3.6e13))
gamma(x::BigFloat) in SpecialFunctions at /Users/gerzaguet/.julia/packages/SpecialFunctions/mFAQ4/src/gamma.jl:578

(testSpecialFunctions) pkg> st
     Project testSpecialFunctions v0.1.0
      Status `~/testSpecialFunctions/Project.toml`
  [276daf66] SpecialFunctions v1.3.0

(testSpecialFunctions) pkg>