i gave birth in Doha as well last year in Al Ahli, however, i was lucky to check out Hamad Hospital since i was asked to meet a specialist to conduct a certain test.
if you must know, al ahli has limited specialists and equipment and most often than not, they recommend you to Hamad Hospital to do the test that they cannot provide.
hamad is practically free while al ahli requires premium fees. both has waiting period/time, which is a major concern for most clients. a lot of clients i saw in al ahli got turned off with it and it happened to me as well. however, waiting in hamad is somehow "accepted" considering you basically pay only 50 riyals for a checkup and including the test you are about to do...it is that cheap.

i chose al ahli because i want my husband and my family to be with me in the hospital.

--> hamad hospital
- cheap, 50 riyals plus you just have to show your
health card that can be secured easily
- clients are referred here to further checkup due to
their specialists and equipment
- husbands not allowed to be with you inside your
room once you deliver
- first come, first served...even if you have an
appointment, so i suggest you always go early
- birth certificate is easy

--> al ahli hospital
- checkups would set you back 300 riyals and 2 free
followups with your OB-GYNE (my OB-GYNE was that
German Doctor mentioned)
- even with appointment, expect to wait at least 30
minutes
- everyone in the family (male and female) are
welcome to be with you in the hospital
- have to submit the request for birth certificate to
Hamad hospital

this is not an exhaustive list, as these were based on my experience only..

hope this helps