Are there any advantages to colonizing planets that are close to each other?

There are no direct effects from colony placement. A colony on the other end of the galaxy will produce the same as one next to your capital.

The main problems from distant colonies are logistical and military:

• You have to pay more influence to claim a system far away from your territory than a close one. Accordingly expanding to the far system, requires a lot of influence spent on the system along the way.

• You either have a far away exclave or a long winded territory, when you go for far colonies, which makes your empire harder to defend.

• Your far colony may be cut off by a non friendly race, thus making military support impossible.

I'm not aware of any strategies which benefit from clustered/distant colonies. Unless you play an adaptable race or set your game with an abundant amount of habitable worlds, the early game mostly consists of colonize whatever you can, due to habitable worlds being quite rare.

Another benefit from colonizing worlds close to your capital is that you have less to worry about protecting your trade flow. If you are within 6 jumps you can even collect all trade directly from your capital as trade hub.

Having a colony far out will require you to setup patrols and can be more easily disrupted.

If all else is equal, a new colony close to your borders is preferable to one further away, for the following reasons.

• Clustered colonies mean your empire has a smaller "footprint," meaning it's easier to defend. Your new colony may even isolate established colonies from the frontier, allowing you to establish a fortress world at a chokepoint into your empire. Your fleets will obviously be able to travel more quickly to clustered colonies (unless you're RPing Protoss (mass gateways)).
• Clustered colonies may share a sector, and therefore a governor. In a high energy empire, this may actually be a detriment, especially if the planets are specialized differently.
• Clustered colonies can be covered by the trade and pirate supression auras of the same starbases, reducing your overall starbase footprint. (Starbase auras also travel by gateway)
• The influence cost of systems increases the further they are from your empire, and disconnected systems increase your empire sprawl, lowering your administrative cap.