There have been many threads on this topic already.
Undergraduate degrees - there are minimal differences as most courses are required to teach similar content to meet the accreditation requirements of professional associations/bodies. You may notice differences in teaching quality, resources and the way in which this material is presented, but overall for bachelor degrees, not much.
Your choice of university definitely should be considered if it's a postgraduate/honours degree.
Obviously as well another consideration is networking opportunities which some unis offer more so than others (e.g. industry placement for applied disciplines such as IT).
I personally don't think the merit of bragging about a UoM or Monash degree is worth isolating yourself from other universities. Consider time/distance from your house, the actual structure of the content on offer etc. before making any decisions. Bragging rights vs. actually being happy.. I know which one i'd pick.
From my perspective - I hold two degrees, a Bachelor degree from La Trobe and a Masters from Monash. IMO I found La Trobe to actually be better in terms of community feel & happiness. I didn't feel that connected with Monash and it was more of a 'in and out' affair. That's not to say you will experience the same, but just providing you with my insight having been to two completely different universities in the span of 5 years.
If you don't like the uni you attend, you can always transfer across. If you get a 90+ ATAR, you CAN choose a uni like La Trobe or Deakin or Swinburne etc... I know many people that have done this and have been completely satisfied with their decision. It's about you and what you want - don't bow to pressure of prestige or parental/peer influence. Do your research, get the facts, understand your options and make a decision. After all, it's your life.
(Sorry, that's the best I can do with the limited context you have provided - if you provide the actual degree/discipline area, perhaps you could get some targeted advice relating to that discipline as to what one you should pick? Also consider that the uni student pool on AN is significantly biased towards UoM/Monash)