Is there any problem if the new hard drive's storage is less than the old one?

The drive only needs to be big enough to hold your operating system, but if the new drive is smaller, you'll have less room for your own data files. For old fashioned hard drives, the highest storage sizes (like 4 TB, terabyte) qualify as "new technology" and tend to be less reliable than smaller drives (like 1 TB). Today, there's no reliable drive, whether old technology or new SSD (solid state drives). There used to be high reliability drives, so the consumer is being played for a fool. Back up your data onto a few different external devices, then you can keep your new, little drive from filling up. Hard drives start misbehaving when they're more than 2/3 full.