In Windows Server We can use Disk Management and create striped,mirrored or raid5 volume on multiple disks also we could put all the disks in a storage pool and create a virtual disk then create volume on top of it in Server Manger. What would the advantage of one of these method over the other?
Using disk management and software based RAID doesn't give you the same performance, reliability or flexibility as Storage spaces in Server 2012 and Server 2012 R2.
Storage Spaces offer features such as dual parity, Storage tiers, Write-back cache, Enhanced support when use with failover clustering.
check out this link:https://technet.microsoft.com/en-us/library/hh831739(v=ws.11).aspx
In short I would always go with Storage Spaces over the traditional software based RAID offerings from Microsoft.
Disk Management is simply that, it manages a set of disks and lets you create volumes, be they RAID or not.
Storage pools is a completely different method of organising storage. Storage pools can also utilise more disk types than "disk management" allows. This link will give you the lowdown on the RAID aspect (one portion) of storage pools
In answer to the "advantages" question directly; I suppose to sum up, storage pools are more flexible. I guess the underlying software RAID will be the same on either (from a practical point of view).
One caveat, bare metal restore is "interesting" for storage pools (you do need to mess about a little more with WMI scripts to attach and detach (you cannot reconfigure storage pools during bare metal restore in the same way regular drives can), I would not recommend keeping system essential material (i.e. stuff you need to get back into windows following a bare metal restore) - you cannot mount system volumes on storage pools anyway but you could easily MOVE data to one.
Of course, if you want to host your OS on a software RAID then you cannot use storage pools for this anyway.